On second-order periodic elliptic operators in divergence form

A.F.M. ter Elst^{1}, Derek W. Robinson^{2} and Adam Sikora^{2}

Abstract

We consider second-order, strongly elliptic, operators with complex coefficients in divergence form on R^{d}. We assume that the coefficients are all periodic with a common period. If the coefficients are continuous we derive Gaussian bounds, with the correct small and large time asymptotic behaviour, on the heat kernel and all its Hölder derivatives. Moreover, we show that the first-order Riesz transforms are bounded on the L_{p}-spaces with p ∈ ⟨1, ∞⟩. Secondly if the coefficients are Hölder continuous we prove that the first-order derivatives of the kernel satisfy good Gaussian bounds. Then we establish that the second-order derivatives exist and satisfy good bounds if, and only if, the coefficients are divergence-free or if, and only if, the second-order Riesz transforms are bounded. Finally if the third-order derivatives exist with good bounds then the coefficients must be constant.

March 1999, revision December 1999

AMS Subject Classification: 35K05, 35B27, 35B40, 47A58.

Home institutions:

1. Department of Mathematics  2. Centre for Mathematics
and Computing Science  and its Applications
Eindhoven University of Technology  School of Mathematical Sciences
P.O. Box 513  Australian National University
5600 MB Eindhoven  Canberra, ACT 0200
The Netherlands  Australia

1 Introduction

We analyze the asymptotic behaviour of complex second-order strongly elliptic operators with periodic coefficients in divergence form acting on L_{2}(R^{d}). We establish that if the coefficients of the operators are continuous then the corresponding semigroups have bounded kernels which, together with their Hölder derivatives, satisfy good large time Gaussian bounds. Moreover, we prove that the Riesz transforms are bounded on the spaces L_{p}(R^{d}) with p ∈ ⟨1, ∞⟩.

Our approach to kernel bounds is through the homogenization theory of periodic systems [BLP] [ZKO]. A novelty of our analysis lies in its use of decomposition theory combined with spectral theory [BJR] [Zhi]. The decomposition, which can be viewed as a partial Fourier transformation, represents the periodic system on R^{d} as a superposition of systems localized on the cell of periodicity and with different boundary conditions. The local systems have discrete spectrum and Zhikov [Zhi] (see [ZKO], Section 2.6) observed that their asymptotic behaviour is determined by the lowest eigenstate. This eigenstate can be calculated by perturbation theory from the purely periodic system. To secondorder the eigenstate coincides with the corresponding state of the homogenized system. This approximation provides a good asymptotic description of the semigroup kernel and higher-order corrections are given by higher-order terms in the perturbation theory.

An essential ingredient to our approach is the approximate Gaussian bounds established by Auscher [Aus] for the semigroup kernels of operators with uniformly continuous coefficients (see also [AMT], [AuT] and [AuQ]). These bounds consist of a Gaussian multiplied by a polynomial in the time variable. Periodicity of the coefficients suffices to remove the growth factor and to give Gaussian bounds for the kernel and its Hölder derivatives for both small and large times. Once one has these kernel bounds one can deduce a local form of boundedness of the Riesz transforms on the L_{p}-spaces from general theory (see [AuT], Section 4.6). Then by relatively straightforward spectral estimates these bounds can be extended to give boundedness of the Riesz transforms on the spaces L_{p}(R^{d}) with p ∈ ⟨1, 2]. But an example of Kenig–Meyers (see [AuT], Chapter 4, Theorem 7) shows that the proof of boundedness for p ∈ [2, ∞⟩ is much more delicate. Nevertheless it can be established by detailed spectral analysis of the localized systems.

Subsequently we examine periodic operators with Hölder continuous coefficients and establish a hierarchy of asymptotic properties. Previously, Avellaneda and Lin [AvL1]

[AvL2] [AvL3] used homogenization theory and De Giorgi estimates to obtain elliptic regularity results for operators with Hölder continuous coefficients. Alexopoulos [Ale1] used similar arguments to establish L_{p}-boundedness of the first-order Riesz transforms of such operators. Avellaneda and Lin considered real systems of elliptic operators, and hence their results encompass the complex case, whilst Alexopoulos considered single real operators. We show that the kernels of periodic operators with Hölder continuous coefficients have a remarkable 1, 2, ∞-structure. The kernels are automatically once-differentiable with the derivatives satisfying good Gaussian bounds; they are twice-differentiable and the second derivatives satisfy good asymptotic bounds if, and only if, the coefficients are divergencefree; existence of the third derivatives with the canonical behaviour implies the coefficients are constant, and then the kernel is infinitely often differentiable. This hierarchy can also be characterized by the boundedness of the Riesz transforms: the first-order transforms are always bounded; boundedness of the second-order transforms is equivalent to the coefficients being divergence-free; boundedness of the third, or higher, order transforms implies the coefficients are constant.

The theory of complex second-order elliptic operators in divergence form differs substantially from the real theory. The complex theory corresponds to a system of two real operators and shares the difficulties encountered with more general systems or with operators in non-divergence form. In 1968 De Giorgi [Gio] gave an example of a system of real operators in divergence form which did not have the regularity features of the Nash–De Giorgi theory of single real operators with measurable coefficients. Several other authors subsequently gave refined examples of this nature (see, for example, [Gia1], Section II.3). Then in 1981 Fabes and Kenig [FaK] constructed a class of real second-order elliptic operators in non-divergence form such that the associated ‘parabolic measures’ are singular with respect to Lebesgue measure. The corresponding semigroups are contractive on L_{∞}(R^{d}) but do not have bounded kernels. These latter examples are surprising since the operators have uniformly continuous coefficients and it is known that Hölder continuity of the coefficients is sufficient to ensure that the kernel exists and is at least twice differentiable (see [Fri], Chapter 9). The construction of the Fabes–Kenig examples was based on ideas of Modica and Mortola [MoM] which in turn used a suggestion of De Giorgi (see also [MMS] [Bau]). The theory of complex divergence form operators presents different structural difficulties. There are no problems in one or two dimensions; the principal regularity results of the Nash–De Giorgi theory are still valid [AMT]. But in 1985 Maz’ya, Nazarov and Plamenevskĭi [MNP] gave counterexamples for the complex theory in higher dimensions. Their ideas have been extended subsequently by various authors [ACT] [Dav] and result in examples of complex divergence form operators which generate continuous semigroups on L_{p}(R^{d}) if, and only if, |1/p − 1/2| is small. In particular the semigroups do not have bounded kernels. These examples require d ≥ 5 and the status of the complex theory for d = 3 and d = 4 is not clear. Nevertheless the reductio ad absurdum argument given in [AMT], Remark 4.24, indicates that smoothness of the coefficients alone is not sufficient to obtain a satisfactory description in terms of bounded kernels. This conclusion is confirmed by our analysis of complex periodic operators.

The principal results of our analysis extend to general systems of complex operators satisfying an appropriate notion of ellipticity. We will extend the results for a complex (scalar) operator to a system of pure second-order operators with complex continuous and periodic coefficients satisfying a strong Gårding inequality. This inequality follows immediately from the strong ellipticity condition, the Legendre condition, used by Avellaneda and Lin [AvL1] [AvL3] in their analysis of real periodic systems. The weaker Legendre– Hadamard condition often used in the analysis of systems (see [Gia2], Section 1.1) does not seem adequate for homogenization or the derivation of asymptotic estimates (see [Zha], [GMT]). The only essential complication in the extension of our arguments to systems satisfying the strong Gårding inequality is in the use of perturbation theory. The discussion of systems requires the technically more complicated degenerate perturbation theory.

In the sequel H denotes the maximal accretive operator on the complex Hilbert space L_{2}(R^{d}) associated with the sectorial form (see, for example, [Kat] [ReS])

h(φ) = ∑_{k,l=1}^{d} (∂_{k}φ, c_{kl}∂_{l}φ)  (1)

where the ∂_{k} = ∂/∂x_{k} denote the usual partial derivatives and the domain of h is D(h) = L_{2;1}(R^{d}) = ⋂_{k=1}^{d} D(∂_{k}). The matrix C = (c_{kl}) of complex-valued coefficients c_{kl} ∈ L_{∞}(R^{d}), which is not necessarily symmetric, is assumed to satisfy the ellipticity condition

ℜC = 2^{−1}(C + C^{∗}) ≥ μI > 0 ,  (2)

in the sense of d × d-matrices over C^{d}, uniformly over R^{d}, where C^{∗} = (c_{kl}^{∗}) with c_{kl}^{∗} = \overline{c_{lk}}. The ellipticity condition ensures that the form h is sectorial with vertex at the origin. The least upper bound μ_{C} of the μ satisfying this condition is called the ellipticity constant.

The operator H generates a continuous holomorphic semigroup S on L_{2}(R^{d}). The action of S is determined by a distribution kernel K. We use the notation L_{p;1}(R^{d}) = ⋂_{k=1}^{d} D(∂_{k}), where ∂_{k} now denotes the partial derivative in L_{p}(R^{d}).

Theorem 1.1 Assume the coefficients c_{kl} of H are continuous and periodic with a common period. Then the kernel K of the semigroup generated by H has the following bounds.

I. There exist b, c > 0 such that

|K_{t}(x; y)| ≤ c G_{b,t}(x − y)  (3)

for all x, y ∈ R^{d} and t > 0 where G_{b,t}(x) = t^{−d/2}e^{−b|x|^{2}t^{−1}}. II. For all ν, τ ∈ ⟨0, 1⟩ and κ > 0 there exist b, c > 0 such that

|K_{t}(x − h; y − k) − K_{t}(x; y)| ≤ c (\frac{|h| + |k|}{|x − y| + t^{1/2}} )^{ν}G_{b,t}(x − y)  (4)

for all x, y, h, k ∈ R^{d} and all t > 0 with |h| + |k| ≤ τ|x − y| + κt^{1/2}. III. If p ∈ ⟨1, ∞⟩ then D(H^{1/2}) = L_{p;1}(R^{d}) and there exist c_{p}, c′_{p} > 0 such that

c_{p} max_{1≤k≤d} ∥∂_{k}φ∥_{p} ≤ ∥H^{1/2}φ∥_{p} ≤ c′_{p} max_{1≤k≤d} ∥∂_{k}φ∥_{p}

for all φ ∈ L_{p;1}(R^{d}).

Although we have stated the kernel bounds (3) and (4) for real positive t they can be extended to all complex t in an open subsector of the sector of holomorphy. A similar remark is true for all subsequent bounds of a similar nature. The passage from real t to complex z = te^{iα} corresponds to replacing H by e^{iα}H and for small α the latter operator is of the same type as H.

The last statement of the theorem gives L_{p}(R^{d})-boundedness of the Riesz transforms R_{k}(H) = ∂_{k}H^{−1/2} for p ∈ ⟨1, ∞⟩. This result was established by Alexopoulos [Ale1] for real operators with Hölder continuous coefficients and Avellaneda and Lin [AvL3] claimed a similar result for systems of such operators. Alexopoulos [Ale2] has, however, pointed out that the latter authors only prove boundedness of R_{k}(H)R_{l}(H^{∗})^{∗} which is a weaker result. Note that the result is optimal in the sense that the Kenig–Meyers example cited above ([AuT], Section IV.2.2) yields a periodic operator with real coefficients which are continuous except at one point in each periodicity cell and for which the transforms are unbounded for all sufficiently large p. Note that for these results one can partially relax the continuity of the coefficients following the arguments of Auscher and Tchamitchian [AuT].

The next theorems summarize our results for operators with Hölder continuous coefficients.

Theorem 1.2 Assume the coefficients c_{kl} of H are Hölder continuous and periodic with a common period. Then K is once-differentiable and there exist b, c > 0 such that

|(∂_{x_{k}}K_{t})(x; y)| + |(∂_{y_{k}}K_{t})(x; y)| ≤ c t^{−1/2}G_{b,t}(x − y)

for all k ∈ {1, …, d}, x, y ∈ R^{d} and all t > 0.

In the fourth condition of the next theorem U denotes the action of translations on the spaces L_{p}(R^{d}).

Theorem 1.3 Assume the coefficients c_{kl} of H are Hölder continuous and periodic with a common period. The following conditions are equivalent.

I. ∑_{k=1}^{d} ∂_{k}c_{kl} = 0, in the sense of distributions, for each l ∈ {1, …, d}.

II. There exist c, ν > 0 such that

|(∂_{x_{k}}K_{t})(x − h; y) − (∂_{x_{k}}K_{t})(x; y)| ≤ c t^{−1/2} (|h|t^{−1/2})^{ν} t^{−d/2}

uniformly for all k ∈ {1, …, d}, x, y, h ∈ R^{d} with |h| ≤ 1 and all t ≥ 1.

III. K is twice-differentiable in the first variable and there exist b, c > 0 such that

|(∂_{x_{k}}∂_{x_{l}}K_{t})(x; y)| ≤ c t^{−1} G_{b,t}(x − y)

for all k, l ∈ {1, …, d}, x, y ∈ R^{d} and t > 0.

IV. There are p ∈ [1, ∞] and c_{p}, ν > 0 such that

∥(I − U(h))∂_{k}S_{t}∥_{p→p} ≤ c_{p} t^{−1/2} (|h|t^{−1/2})^{ν}  (5)

uniformly for all k ∈ {1, …, d}, all |h| ≤ 1 and all t ≥ 1.

V. If p ∈ ⟨1, ∞⟩ then D(H) = L_{p;2}(R^{d}) and there exist c_{p}, c′_{p} > 0 such that

c_{p} max_{1≤k,l≤d} ∥∂_{k}∂_{l}φ∥_{p} ≤ ∥Hφ∥_{p} ≤ c′_{p} max_{1≤k,l≤d} ∥∂_{k}∂_{l}φ∥_{p}

for all φ ∈ L_{p;2}(R^{d}).

The next result shows that further regularity occurs for all t only in the simple case that the coefficients are constant. Then of course the kernel is Gaussian and the Riesz transforms of all orders are bounded.

Theorem 1.4 Assume the coefficients c_{kl} of H are Hölder continuous and periodic with a common period. The following conditions are equivalent.

I. The coefficients c_{kl} are constant.

II. K is twice-differentiable in the first variable and there exist c, ν > 0 such that

|(∂_{x_{k}}∂_{x_{l}}K_{t})(x − h; y) − (∂_{x_{k}}∂_{x_{l}}K_{t})(x; y)| ≤ c t^{−1} (|h|t^{−1/2})^{ν} t^{−d/2}

uniformly for all k, l ∈ {1, …, d}, x, y, h ∈ R^{d} with |h| ≤ 1 and all t ≥ 1.

III. There are p ∈ [1, ∞] and c_{p}, ν > 0 such that S_{t}L_{p}(R^{d}) ⊂ L_{p;2}(R^{d}) and

∥(I − U(h))∂_{k}∂_{l}S_{t}∥_{p→p} ≤ c_{p} t^{−1} (|h|t^{−1/2})^{ν}

uniformly for all k, l ∈ {1, …, d}, all |h| ≤ 1 and all t ≥ 1.

IV. If p ∈ ⟨1, ∞⟩ then D(H^{3/2}) = L_{p;3}(R^{d}) and there exist c_{p}, c′_{p} > 0 such that

c_{p} max_{1≤k,l,m≤d} ∥∂_{k}∂_{l}∂_{m}φ∥_{p} ≤ ∥H^{3/2}φ∥_{p} ≤ c′_{p} max_{1≤k,l,m≤d} ∥∂_{k}∂_{l}∂_{m}φ∥_{p}

for all φ ∈ L_{p;3}(R^{d}).

Although the regularity properties in these theorems are stated in terms of the first variable of the kernel one can readily draw similar conclusions for the second variable by symmetry. Interchange of the variables corresponds, by duality, to replacement of C by its transpose C^{t}, i.e., the matrix with coefficients c_{kl}^{t} = c_{lk}. For example, under the conditions of Theorem 1.3 the derivative with respect to the second variable is Hölder continuous with the canonical behaviour if, and only if, ∑_{k=1}^{d} ∂_{k}c_{kl}^{t} = ∑_{k=1}^{d} ∂_{k}c_{lk} = 0 as distributions for each l ∈ {1, …, d}. In fact one can deduce joint regularity properties from separate regularity properties by using the semigroup property

K_{s+t}(x; y) = ∫_{R^{d}} dz K_{s}(x; z)K_{t}(z; y) .

For example, it follows in Theorem 1.2 that K is once-differentiable jointly with respect to both variables.

In Section 2 we summarize some basic elements of decomposition theory and homogenization theory for operators on L_{2}(R^{d}) with periodic coefficients and derive the essential spectral estimates. In Section 3 we prove Theorem 1.1. This follows by establishing that the kernel of the homogenized operator gives the best asymptotic approximation to K. We also deduce some results on first-order corrections to the asymptotic series for the kernel. In Section 4 we prove Theorems 1.2, 1.3 and 1.4. The conditions of Theorem 1.3 correspond to the first-order corrector, in the sense of homogenization theory, vanishing. The conditions of Theorem 1.4 correspond to the first and second order correctors vanishing, and then the correctors of all orders vanish.

Finally note that Property I of Theorem 1.3 allows one to express H as a pure secondorder operator in non-divergent form. Hence it suggests examining the possible validity of the other properties in the theorem for general operators of this type. But it follows from [Fri], Chapter 9, that each operator H_{n} = − ∑_{k,l=1}^{d} c_{kl}∂_{k}∂_{l} in non-divergent form with strongly elliptic, Hölder continuous, coefficients generates a semigroup with a bounded integrable kernel. Although this kernel satisfies Gaussian bounds for small t it is unlikely that it would have good large t behaviour unless H_{n} satisfies some additional dissipativity condition. Nevertheless, Property V of the theorem, the boundedness of the second-order Riesz transforms R_{k,l}(H_{n}) = ∂_{k}∂_{l}H_{n}^{−1} can be verified for non-divergent operators with real coefficients. This is discussed in further detail in Section 4.

In Section 5 we briefly discuss systems of complex operators. We will point out the changes needed to extend our results to such systems.

2 Periodic systems

The proof of Theorem 1.1 relies on the a priori estimates of Auscher and some standard arguments of the theory of elliptic operators in combination with various Hilbert space properties of periodic systems. We develop the latter in this section. Throughout the section we suppose that H is the maximal accretive operator associated with the sectorial form (1) with the complex measurable coefficients satisfying the ellipticity assumption (2). We further assume that the coefficients are all periodic with a common period. Then, for simplicity, we choose units such that

c_{kl}(x + n) = c_{kl}(x) ,  (6)

for all k, l ∈ {1, …, d}, x ∈ R^{d} and n ∈ Z^{d}. No continuity of the coefficients is required in this section.

2.1 Decomposition theory

Decomposition theory for operators with periodic coefficients has a long history dating back to Bloch [Blo]. We follow the description in [BJR] which is based on the Zak formalism commonly used in wavelet theory (see, for example [Dau]) although we then pass to the unitarily equivalent Bloch prescription.

If U denotes the unitary action of R^{d} by left translations on L_{2}(R^{d}), i.e.,

(U(y)φ)(x) = φ(x − y)

for all φ ∈ L_{2}(R^{d}) and y ∈ R^{d}, then U(x)D(h) = D(h) for all x ∈ R^{d} and the periodicity (6) of the coefficients gives the invariance property h(U(n)φ) = h(φ) for all φ ∈ D(h) and all n ∈ Z^{d}. Hence U(n)D(H) = D(H) and U(n)H = HU(n). Moreover,

U(n)S_{t} = S_{t}U(n)

on L_{2}(R^{d}) for all n ∈ Z^{d} and all t > 0. Next we introduce versions of H and S on L_{2}(I^{d}) where I = [0, 1].

Let ∂_{k}, for k ∈ {1, …, d}, denote the partial derivatives on L_{2}(I^{d}) corresponding to periodic boundary conditions, φ(u_{1}, …, 0, …, u_{d}) = φ(u_{1}, …, 1, …, u_{d}) for almost every u ∈ I^{d}, where the 0 and 1 are in the k-th position. Secondly, for all θ ∈ [−π, π]^{d} define H_{θ} as the maximal accretive operator on L_{2}(I^{d}) associated with the sectorial form

h_{θ}(φ) = ∑_{k,l=1}^{d}((∂_{k} + iθ_{k})φ, c_{kl}(∂_{l} + iθ_{l})φ)  (7)

where D(h_{θ}) = ⋂_{k=1}^{d} D(∂_{k}) = L_{2;1}(I^{d}).

The H_{θ} are unitarily equivalent to versions H̃_{θ} of H with θ-periodic boundary conditions. The unitarily equivalent description was used in [BJR], Section 2, and is related to the present description as follows. Let ∂_{k}^{θ}, for k ∈ {1, …, d}, denote the partial derivatives on L_{2}(I^{d}) corresponding to the θ-periodic boundary conditions, φ(u_{1}, …, 0, …, u_{d}) = e^{iθ_{k}}φ(u_{1}, …, 1, …, u_{d}) for all k ∈ {1, …, d} and almost every u ∈ I^{d}, where the 0 and 1 are in the k-th position. In particular ∂_{k}^{0} coincides with the periodic operator ∂_{k}. Further let V (θ) be the unitary operator on L_{2}(I^{d}) defined by (V (θ)φ)(u) = e^{iu·θ}φ(u) for u ∈ I^{d}. If φ ∈ ⋂_{k=1}^{d} D(∂_{k}) then V (θ)φ ∈ ⋂_{k=1}^{d} D(∂_{k}^{θ}) and ∂_{k}^{θ}V (θ)φ = V (θ)(∂_{k} + iθ_{k})φ. Consequently,

h_{θ}(φ) = ∑_{k,l=1}^{d}(∂_{k}^{θ}V (θ)φ, c_{kl}∂_{l}^{θ}V (θ)φ) .

Hence H_{θ} = V (θ)^{∗}H̃_{θ}V (θ) where the H̃_{θ} are the operators used in [BJR]. Formally H̃_{θ} = − ∑_{k,l=1}^{d} ∂_{k}^{θ}c_{kl}∂_{l}^{θ}.

One has a (Bloch) decomposition of L_{2}(R^{d}),

L_{2}(R^{d}) ≅ (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ L_{2}(I^{d})_{θ} ,

as a direct integral of copies of L_{2}(I^{d}) indexed by θ ∈ [−π, π]^{d} (for a description of the formalism of integral decompositions see, for example, [Dix], Chapter II, or [BrR], Chapter IV). The isomorphism is given as follows. Define Φ: L_{2}(R^{d}) → (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ L_{2}(I^{d})_{θ} by (Φ(φ))(θ) = φ_{θ}, where

φ_{θ}(u) = ∑_{n∈Z^{d}} e^{i(n−u)·θ}φ(u − n)

for all φ ∈ L_{2}(R^{d}), almost every θ ∈ [−π, π]^{d} and almost every u ∈ I^{d}. By Fubini’s and Parseval’s theorems one verifies that Φ is a unitary operator. The inverse of Φ is given by

(Φ^{−1}((φ_{θ})_{θ∈[−π,π]^{d}}))(u − n) = (2π)^{−d} ∫_{[−π,π]^{d}} dθ e^{−i(n−u)·θ}φ_{θ}(u)

for all (φ_{θ})_{θ∈[−π,π]^{d}} ∈ (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ L_{2}(I^{d})_{θ}, n ∈ Z^{d} and almost every u ∈ I^{d}.

One can also introduce an alternative (Zak) decomposition with components φ̃_{θ} given by

φ̃_{θ}(u) = ∑_{n∈Z^{d}} e^{in·θ}φ(u − n) = (V (θ)φ_{θ})(u)

for all φ ∈ L_{2}(R^{d}), almost every θ ∈ [−π, π]^{d} and almost every u ∈ I^{d}. This latter decomposition was used in [BJR] and will be used on occasions in the sequel. The corresponding unitary operator will be denoted by Φ̃.

It follows from [BJR] that the sectorial form h, the operator H and the semigroup S can be decomposed in a similar manner. Explicitly, if Φ(φ) = (φ_{θ})_{θ∈[−π,π]^{d}} then

h(φ) = (2π)^{−d} ∫_{[−π,π]^{d}} dθ h_{θ}(φ_{θ})

and

ΦHΦ^{−1} = (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ H_{θ}

in the sense of direct integral decompositions of closed sectorial forms. Similarly

ΦS_{t}Φ^{−1} = (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ S_{t}^{θ}  (8)

where S^{θ} denotes the semigroup generated by H_{θ} on L_{2}(I^{d}) (see [BJR], Theorem 2.3). If S has a bounded integrable kernel then the semigroup decomposition gives a similar kernel decomposition. We will use this fact extensively in Section 3.

If A is any Z^{d}-invariant bounded operator on L_{2}(R^{d}), i.e., a bounded operator such that AU(n) = U(n)A for all n ∈ Z^{d}, then ΦAΦ^{−1} has a decomposition

ΦAΦ^{−1} = (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ A_{θ}  (9)

where the A_{θ} are bounded operators on L_{2}(I^{d}). It is important in the following to be able to estimate the norms of decomposable operators on L_{2}(R^{d}), or more generally on L_{p}(R^{d}), in terms of the comparable norms of their components on L_{2}(I^{d}), or L_{p}(I^{d}). Estimates of this type are particularly easy for L_{1} → L_{∞}-crossnorms. Note that in the subsequent discussion the norms and crossnorms on A are computed on R^{d}, whilst the norms and crossnorms on A_{θ} are on I^{d}.

Lemma 2.1 Let A be a bounded operator on L_{2}(R^{d}) with a decomposition Φ^{−1}AΦ = (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ A_{θ} on the direct integral space (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ L_{2}(I^{d})_{θ}. Then

∥A∥_{1→∞} ≤ (2π)^{−d} ∫_{[−π,π]^{d}} dθ ∥A_{θ}∥_{1→∞} .

Proof If φ ∈ C_{c}(R^{d}) then ∥φ_{θ}∥_{1} ≤ ∥φ∥_{1} for all θ ∈ [−π, π]^{d}. Therefore |(ψ, Aφ)| = |(2π)^{−d} ∫_{[−π,π]^{d}} dθ (ψ_{θ}, A_{θ}φ_{θ})| ≤ |(2π)^{−d} ∫_{[−π,π]^{d}} dθ ∥ψ_{θ}∥_{1} ∥A_{θ}∥_{1→∞} ∥φ_{θ}∥_{1} ≤ ∥ψ∥_{1} ∥φ∥_{1} (2π)^{−d} ∫_{[−π,π]^{d}} dθ ∥A_{θ}∥_{1→∞}, where φ_{θ} = (Φ(φ))(θ) and ψ_{θ} = (Φ(ψ))(θ), from which the lemma follows. Note that the crossnorms can be infinite. □

The computation of L_{2} → L_{2} norms is also straightforward since the L_{2}(R^{d})-norm of a Z^{d}-invariant operator A is readily calculated in terms of the L_{2}(I^{d})-norms of its components A_{θ}. The Plancherel formula gives

∥A∥_{2→2} = ess sup_{θ∈[−π,π]^{d}} ∥A_{θ}∥_{2→2}  (10)

but there is no comparable direct link between the L_{p}-norms if p ≠ 2. Nevertheless there is one simple criterion for boundedness on the L_{p}-spaces for operators with components which are periodic, i.e., A_{(θ_{1},...,−π,...,θ_{d})} = A_{(θ_{1},...,π,...,θ_{d})} for all k ∈ {1, …, d} and almost every θ ∈ [−π, π]^{d}, where the −π and π are in the k-th position.

Lemma 2.2

I. Let A be a bounded operator on L_{2}(R^{d}) with a Bloch decomposition (9) on the direct integral space (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ L_{2}(I^{d})_{θ}. Let N > d and p ∈ [1, ∞]. Assume the components are periodic and that the map θ ↦ A_{θ} extends to an N-times continuous differentiable map from [−π, π]^{d} into L(L_{p}(I^{d})). Then A is bounded on L_{p}(R^{d}). Moreover,

∥A∥_{p→p} ≤ c sup_{θ∈[−π,π]^{d}} sup_{|α|≤N} ∥\frac{∂^{α}}{∂θ^{α}} A_{θ}∥_{p→p} ,

where c > 0 is a constant which depends only on N and d.

II. Similar results are valid for the Zak decomposition instead of the Bloch decomposition.

Proof We prove only Statement I, since the other can be proved similarly. Define A_{n} on L_{p}(I^{d}) by

A_{n} = (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ e^{−in·θ}A_{θ}

for all n ∈ Z^{d}. Then

(Aφ)(u − n) = ∑_{m∈Z^{d}}(A_{m}U(n − m)φ)(u)

for all u ∈ I^{d}, n ∈ Z^{d} and φ ∈ C_{c}(R^{d}). Hence

∥Aφ∥_{p}^{p} ≤ ∑_{n∈Z^{d}} | ∑_{m∈Z^{d}} ∥A_{m}∥_{p→p}∥U(n − m)φ∥_{p}|^{p}

where the norm on the left is over R^{d} and the norms on the right over I^{d}. But

m^{α}A_{m} = (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ e^{−in·θ}( − i\frac{∂^{α}}{∂θ^{α}} )A_{θ}

where we have used the periodicity of θ ↦ A_{θ} to integrate by parts. Consequently one has bounds

∥A_{m}∥_{p→p} ≤ c_{N}(1 + |m|)^{−N}

uniformly for all m ∈ Z^{d}. Therefore

∥Aφ∥_{p}^{p} ≤ ∑_{n∈Z^{d}} | ∑_{m∈Z^{d}} c_{N}(1 + |m|)^{−N}∥U(n − m)φ∥_{p}|^{p} ≤ (c′_{N})^{p}∥φ∥_{p}^{p}

since N > d where the last estimate follows from the discrete form of Minkowski’s inequality ∥a ∗ b∥_{p} ≤ ∥a∥_{1}∥b∥_{p}. □

The second criterion for L_{p}(R^{d})-boundedness of a decomposable operators is more restricted and more delicate. It applies to operators which are Z^{d}-invariant multipliers, i.e., there is an a ∈ L_{∞}([−π, π]^{d}) such that A_{θ} = a(θ)I for almost all θ ∈ [−π, π]^{d}.

Proposition 2.3 Let A be a Z^{d}-invariant multiplier with multiplication function a and N the smallest integer strictly larger than d/2. Suppose supp a ⊂ ⟨−π, π⟩^{d} and that a is N-times differentiable on R^{d}\{0} with derivatives satisfying bounds

|(∂_{θ}^{α}a)(θ)| ≤ c_{α} |θ|^{−|α|}

for all multi-indices with |α| ≤ N. Then ∥A∥_{p→p} < ∞ on L_{p}(R^{d}) for all p ∈ ⟨1, ∞⟩. Moreover, A is of weak type (1, 1).

Proof Since A and A^{∗} are bounded on L_{2}(R^{d}) it suffices to prove that they are of weak type (1, 1). This is achieved by the standard methods of singular integration theory.

First, the action of A on L_{p}(R^{d}) is given by convolution

(Aφ)(u − n) = ∑_{m∈Z^{d}} K(n − m)φ(u − m)

with a function K,

K(n) = (2π)^{−d} ∫_{[−π,π]^{d}} dθ e^{−in.θ}a(θ)

over Z^{d} which automatically satisfies |K(n)| ≤ ∥a∥_{∞} for all n ∈ Z^{d}.

Secondly, one establishes that there is a c > 0 such that

∑_{n∈Z^{d}:|n|≥2|m|} |K(n − m) − K(n)| < c  (11)

uniformly for all m ∈ Z^{d}. The proof follows by a straightforward modification of the argument used to prove the second part of Proposition 2 in Section 4.4 of Chapter VI in [Ste1].

Thirdly, consider the operator Ã defined by

(Ãf)(n) = ∑_{m∈Z^{d}} K(n − m)f(m)

for f ∈ l_{2}(Z^{d}). Then Ã considered as an operator on l_{2}(Z^{d}) satisfies ∥Ãf∥_{2} ≤ ∥a∥_{∞}∥f∥_{2} and it follows from the estimates (11) that Ã has properties similar to a Calderón–Zygmund operator. Therefore it of weak type (1, 1) over Z^{d}, i.e., there is a C > 0 such that

∑_{n∈Z^{d}} χ_{{n:|(_{A}f)(n)|>λ}}(n) ≤ C λ^{−1} ∑_{n∈Z^{d}} |f(n)|

for all f ∈ l_{1}(Z^{d}) ∩ l_{2}(Z^{d}) and λ > 0 where χ_{Ω} is the characteristic function of the subset Ω ⊂ Z^{d}. Precisely, the weak type (1, 1) inequality follows from the proof of Theorem I.3 of [Ste2], if one uses Corollary III.2.3 of [CoW] instead of Theorem I.2 of [Ste2]. Therefore setting f(n) = φ(u − n) with φ ∈ L_{1}(R^{d}) ∩ L_{2}(R^{d}) and u ∈ I^{d} one obtains

∑_{n∈Z^{d}} χ_{{(u,n):|(Aφ)(u−n)|>λ}}(u, n) ≤ C λ^{−1} ∑_{n∈Z^{d}} |φ(u − n)|

for almost all u ∈ I^{d} where χ_{Ω} is now the characteristic function of the subset Ω ⊂ I^{d} × Z^{d}. Integrating over I^{d} establishes that A is of weak type (1, 1) as an operator over R^{d} and completes the proof of the proposition. □

Lemma 2.2 and Proposition 2.3 will be used in Subsection 3.4 to prove boundedness of Riesz transforms.

2.2 Spectral estimates

In this subsection we derive some basic spectral properties of the operators H_{θ} associated with the sectorial forms (7). It then follows that for large t the semigroup is determined by the small θ components S^{θ} in the periodic decomposition. Moreover, for small |θ| the large t behaviour of the S^{θ} is determined by the lowest eigenvalue and its eigenprojector.

It follows immediately from the definition (7) that the family h_{θ} of sectorial forms extends to a (multi-variable) holomorphic family of type (a), with θ ∈ C^{d} (see [Kat], VII- §4.2). We denote the extension by h_{θ} and the associated m-sectorial operator by H_{θ}. Then the family H_{θ}, θ ∈ C^{d}, is a (multi-variable) holomorphic family of operators of type (B) by [Kat], Theorem VII-4.2. (Although we have a d-parameter extension of the formalism of Kato this does not affect any of the results we adopt from the standard theory.)

Each H_{θ} has a compact resolvent. This follows because H_{0} has compact resolvent by [BJR], Lemma 2.1, and the family H_{θ} are a holomorphic family of type (B). Then every H_{θ} has compact resolvent for all θ ∈ C^{d} (see [Kat], Theorem 4.3).

For all δ ∈ [0, π] set

B_{δ} = {θ ∈ [−π, π]^{d} : max_{k∈{1,...,d}} |θ_{k}| ≤ δ} .

Then B_{δ} is closed in [−π, π]^{d}.

Lemma 2.4 If κ ∈ [0, 1] then

Re h_{θ}(φ) ≥ μ_{C} (1 − κ)∥∇φ∥_{2}^{2} + μ_{C} |θ|^{2} ∥φ∥_{2}^{2}

for all φ ∈ L_{2;1}(I^{d}) and θ ∈ B_{κπ}.

Proof Let (c_{n})_{n∈Z} ∈ C_{c}(Z^{d}) and consider the trigonometric polynomial φ given by φ(u) = ∑_{n∈Z^{d}} c_{n} e^{2πin·u}. Then

Re h_{θ}(φ) = Re ∫_{I^{d}} du ∑_{k,l=1}^{d} \overline{((∂_{k} + i θ_{k})φ)(u)} c_{kl}(u) ((∂_{l} + i θ_{l})φ)(u)

≥ μ_{C} ∫_{I^{d}} du ∑_{k=1}^{d} |((∂_{k} + i θ_{k})φ)(u)|^{2} = μ_{C} ∑_{k=1}^{d} ∑_{n∈Z^{d}} |c_{n}|^{2} (2π n_{k} + θ_{k})^{2} .

But if m ∈ Z and ν ∈ R with |ν| ≤ κ π then (2πm + ν)^{2} ≥ 4π^{2}m^{2}(1 − κ) + ν^{2}. Hence

Re h_{θ}(φ) ≥ μ_{C} ∑_{k=1}^{d} ∑_{n∈Z^{d}} |c_{n}|^{2}(4π^{2} n_{k}^{2}(1 − κ) + |θ_{k}|^{2}) = μ_{C}(1 − δ) ∑_{k=1}^{d} ∥∂_{k}φ∥_{2}^{2} + μ_{C} |θ|^{2} ∥φ∥_{2}^{2} .

Since the trigonometric polynomials are dense in L_{2;1}(I^{d}) the lemma follows by a limiting argument. □

The lemma has four immediate corollaries.

Corollary 2.5 If θ ∈ [−π, π]^{d} then ∥S_{t}^{θ}∥_{2→2} ≤ e^{−μ_{C}|θ|^{2}t} for all t > 0.

Corollary 2.6 If θ ∈ [−π, π]^{d} and λ is an eigenvalue of H_{θ} then Re λ ≥ μ_{C} |θ|^{2}.

These statements follow immediately since Re h_{θ}(φ) ≥ μ_{C} |θ|^{2} ∥φ∥_{2}^{2} for all φ ∈ L_{2;1}(I^{d}) and θ ∈ [−π, π]^{d} by taking κ = 1 in Lemma 2.4.

Corollary 2.7 There exists a c > 0 such that

Re h_{θ}(φ) ≥ c Re h_{0}(φ)

for all φ ∈ L_{2;1}(I^{d}) and θ ∈ [−π, π]^{d}.

Proof Let ∥C∥ = sup_{u∈I^{d}} ∥(c_{kl}(u))∥_{2→2}. Then it follows from the definition of h_{θ} that

|h_{0}(φ) − h_{θ}(φ)| ≤ 2∥C∥ ∥∇φ∥_{2} |θ| ∥φ∥_{2} + ∥C∥ |θ|^{2} ∥φ∥_{2}^{2}
≤ ε μ_{C} ∥∇φ∥_{2}^{2} + ε^{−1} μ_{C}^{−1}∥C∥^{2} |θ|^{2} ∥φ∥_{2}^{2} + ∥C∥ |θ|^{2} ∥φ∥_{2}^{2}
≤ ε Re h_{0}(φ) + μ_{C}^{−1}(ε^{−1} μ_{C}^{−1} ∥C∥^{2} + ∥C∥) Re h_{θ}(φ)

for all ε > 0 and φ ∈ L_{2;1}(I^{d}), where, in the last step, we have used Lemma 2.4. The corollary follows by taking ε = 1/2. □

Clearly the operator H_{0} and its adjoint H_{0}^{∗} have eigenvalue 0 with eigenfunction 11, the identity function. But it follows from the lemma that this eigenvalue is simple, i.e., it has multiplicity one.

Corollary 2.8 If λ ≠ 0 is an eigenvalue of H_{0} then Re λ > 0. Moreover, 0 is a simple eigenvalue of H_{0}.

Proof If ψ is a non-constant eigenfunction of the operator H_{0} with eigenvalue λ then Re λ ∥ψ∥_{2}^{2} = Re h_{0}(ψ) ≥ μ_{C}∥∇ψ∥_{2}^{2} > 0. This implies both statements. □

For all δ > 0 set

B_{δ}^{C} = {θ ∈ C^{d} : max_{k∈{1,...,d}} |θ_{k}| < δ} .

Then B_{δ}^{C} is open in C^{d}.

Proposition 2.9 There exist ε_{0} > 0, θ_{0} ∈ ⟨0, π⟩ and a holomorphic function λ_{0}: B_{θ_{0}}^{C} → C such that λ_{0}(0) = 0 and λ_{0}(θ) is a simple eigenvalue of H_{θ} for all θ ∈ B_{θ_{0}}^{C}. Moreover, λ_{0}(θ) is the unique eigenvalue λ of H_{θ} with |λ| ≤ ε_{0} for all θ ∈ B_{θ_{0}}^{C}.

Proof The first part follows by Corollary 2.8 and the proof of Theorem VII-1.7 in [Kat].

By Corollary 2.8 there exists an ε_{0} > 0 such that 0 is the unique eigenvalue λ of H_{0} with |λ| ≤ ε_{0}. Moreover, the dimension of the eigenspace equals one. But these properties are stable under perturbation. It then follows from the proof of [Kat], Theorem VII-1.7, that for all sufficiently small |θ| the eigenvalue λ_{0}(θ) is the unique eigenvalue λ of H_{θ} with |λ| < ε_{0}. □

Let ε_{0}, θ_{0} and λ_{0} be as in Proposition 2.9. For θ ∈ B_{θ_{0}}^{C} define the projection

P_{0}(θ) = (2πi)^{−1} ∫_{Γ_{ε_{0}}} dλ (λI − H_{θ})^{−1}  (12)

where Γ_{ε_{0}} = {z ∈ C : |z| = ε_{0}}. Then the map θ ↦ P_{0}(θ) is holomorphic near θ = 0. This again follows by the proof of [Kat], Theorem VII-1.7. Therefore we may assume that the map θ ↦ P_{0}(θ) is holomorphic on B_{θ_{0}}^{C} and that

∥P_{0}(θ) − P_{0}(0)∥_{2→2} ≤ 1/2

for all θ ∈ B_{θ_{0}}^{C}. Note that P_{0}(0) is the orthogonal projection onto the constant functions.

Define φ_{0}: B_{θ_{0}}^{C} → L_{2}(I^{d}) by φ_{0}(θ) = P_{0}(θ)11. Then ∥φ_{0}(θ)−11∥_{2} = ∥(P_{0}(θ)−P_{0}(0))11∥_{2} ≤ 1/2, so φ_{0}(θ) ≠ 0. Since P_{0}(θ) projects onto the one-dimensional eigenspace corresponding to the eigenvalue λ_{0}(θ) it follows that φ_{0}(θ) is an eigenfunction of H_{θ} with eigenvalue λ_{0}(θ). Obviously the map ↦ φ_{0} is holomorphic from B_{θ_{0}}^{C} into L_{2}(I^{d}). But φ_{0}(θ) ∈ D(H_{θ}) ⊆ D(h_{θ}) = L_{2;1}(I^{d}) for each θ ∈ B_{θ_{0}}^{C}. We next show that the map θ ↦ φ_{0}(θ) is also holomorphic from B_{θ_{0}}^{C} into L_{2;1}(I^{d}).

Lemma 2.10 For all k ∈ {1, …, d} the map θ ↦ ∂_{k}φ_{0}(θ) is holomorphic from B_{θ_{0}}^{C} into L_{2}(I^{d}).

Proof Obviously the map θ ↦ (ψ, ∂_{k}φ_{0}(θ)) = −(∂_{k}ψ, φ_{0}(θ)) is holomorphic on B_{θ_{0}}^{C} for all ψ ∈ L_{2;1}(I^{d}) and since L_{2;1}(I^{d}) is dense in L_{2}(I^{d}) it suffices to show that the map θ ↦ ∂_{k}φ_{0}(θ) from B_{θ_{0}}^{C} into L_{2}(I^{d}) is locally bounded (see [Kat], Remark III-1.38). But

∥∂_{k}φ_{0}(θ)∥_{2} = ∥e^{λ_{0}(θ)}∂_{k}S_{1}^{θ}φ_{0}(θ)∥_{2} ≤ e^{Re λ_{0}(θ)}∥∂_{k}S_{1}^{θ}∥_{2→2}∥φ_{0}(θ)∥_{2}

so it reduces the problem to showing that θ ↦ ∥∂_{k}S_{1}^{θ}∥_{2→2} is locally bounded. Since

Re(φ, H_{θ}φ) ≥ 2^{−1}μ_{C}∥∇φ∥_{2}^{2} − ∥C∥ |θ|^{2}(1 + 2μ_{C}^{−1}∥C∥)∥φ∥_{2}^{2}

for all φ ∈ D(H_{θ}) and θ ∈ C^{d} it follows by a straightforward argument that the map θ ↦ ∥∂_{k}S_{1}^{θ}∥_{2→2} is locally bounded on C^{d} (cf. the proof of [Rob], Lemma III.4.4). □

Proposition 2.11 There exists a μ > 0 such that

Re(φ, H_{θ}φ) ≥ μ ∥φ∥_{2}^{2}

for all θ ∈ B_{2^{−1}θ_{0}} and φ ∈ (I − P_{0}(θ))(D(H_{θ})).

Proof Let θ ∈ B_{2^{−1}θ_{0}} and φ ∈ (I − P_{0}(θ))(D(H_{θ})). Then φ ∈ D(H_{θ}) ⊆ D(h_{θ}) = L_{2;1}(I^{d}) and

Re(φ, H_{θ}φ) ≥ c Re h_{0}(φ) = c Re h_{0}((I − P_{0}(0))φ)

where c is the constant of Corollary 2.7 and we have used H_{0} P_{0}(0) = H_{0}^{∗} P_{0}(0) = 0. But the self-adjoint operator Re H_{0} associated with the quadratic form Re h_{0} has a discrete spectrum with simple lowest eigenvalue 0 and eigenfunction 11 (see Corollary 2.8). So if λ_{1} = min(σ(Re H_{0})\{0}) then

Re(φ, H_{θ}φ) ≥ c λ_{1}∥(I − P_{0}(0))φ∥_{2}^{2} = c λ_{1}(∥φ∥_{2}^{2} − (φ, P_{0}(0)φ)) .

Since ∥P_{0}(θ) − P_{0}(0)∥_{2→2} ≤ 2^{−1} for all θ ∈ B_{θ_{0}}^{C} one has

Re(φ, H_{θ}φ) ≥ c λ_{1}(2^{−1} ∥φ∥_{2}^{2} − Re(φ, P_{0}(θ)φ)) = 2^{−1} c λ_{1} ∥φ∥_{2}^{2}

for all θ ∈ B_{2^{−1}θ_{0}} and φ ∈ (I − P_{0}(θ))(D(H_{θ})). □

Proposition 2.12 There exists a μ > 0 such that

∥S_{t}^{θ}(I − P_{0}(θ))φ∥_{2} ≤ e^{−μt}∥(I − P_{0}(θ))φ∥_{2}

uniformly for all t > 0, φ ∈ L_{2}(I^{d}) and θ ∈ B_{2^{−1}θ_{0}}.

Proof Let μ > 0 be as in Proposition 2.11 and θ ∈ B_{2^{−1}θ_{0}}. The subspace H = (I − P_{0}(θ))(L_{2}(I^{d})) is the orthogonal complement of the one-dimensional subspace P_{0}(θ)^{∗}L_{2}(I^{d}). Since the latter subspace is invariant under the adjoint of S^{θ} it follows that H is invariant under S^{θ}. Moreover, the restriction T of S^{θ} to H is a continuous, holomorphic, semigroup with generator H_{θ}|_{H∩D(H_{θ})}. But then it follows from Proposition 2.11 that Re(φ, H_{θ}φ) ≥ μ ∥φ∥_{2}^{2} for all φ ∈ H ∩ D(H_{θ}). So ∥T_{t}∥_{H→H} ≤ e^{−μt} for all t > 0, as required. □

2.3 Homogenization

The asymptotic behaviour of periodic operators and the associated semigroups is often determined by the related, homogenized system (see [BLP] or [ZKO]). Next we define the homogenized operator corresponding to H and its decomposition. To this end we need a boundedness result for the Riesz transforms associated with H_{0}.

First, the self-adjoint operator Re H_{0} associated with the quadratic form Re h_{0} has a discrete spectrum with simple lowest eigenvalue 0 and eigenfunction 11. Hence by [Kat], Theorem III-6.17, the operator Re H_{0} leaves the orthogonal complement L_{2}^{⊥}(I^{d}) = (I − P_{0}(0))(L_{2}(I^{d})) of the constant functions in L_{2}(I^{d}) invariant and the restriction of Re H_{0} to the space L_{2}^{⊥}(I^{d}) is invertible. With a slight abuse of notation we denote this inverse by (Re H_{0})^{−1}. In this section we shall use similar notation. Then the operator (Re H_{0})^{−1/2} = ((Re H_{0})^{−1})^{1/2} is also bounded on L_{2}^{⊥}(I^{d}). But since

Re h_{0}(φ) ≥ μ_{C}∥∇φ∥_{2}^{2}

it follows that ∂_{k}(Re H_{0})^{−1/2} is bounded (has bounded closure) and (Re H_{0})^{−1/2}∂_{l} extends to a bounded operator, on L_{2}^{⊥}(I^{d}). Hence ∂_{k}(Re H_{0})^{−1}∂_{l} extends to a bounded operator, on L_{2}^{⊥}(I^{d}), for all k, l ∈ {1, …, d}. This last result has an analogue with Re H_{0} replaced by H_{0}. Note that this does not require any continuity of the coefficients.

Lemma 2.13 The inverse H_{0}^{−1} is bounded on L_{2}^{⊥}(I^{d}) and for all k, l ∈ {1, …, d} the forms

x_{l}(φ) = (φ, H_{0}^{−1}∂_{l}φ) , x_{kl}(φ) = (∂_{k}φ, H_{0}^{−1}∂_{l}φ) ,

with D(x_{l}) = D(x_{kl}) = L_{2;1}(I^{d}), have bounded extensions to L_{2}(I^{d}).

If X_{l} and X_{kl} denote the bounded operators associated with the bounded forms x_{l} and x_{kl} on L_{2}(I^{d}) then X_{l} and X_{kl} map L_{2}(I^{d}) into L_{2}^{⊥}(I^{d}). Moreover, X_{l}L_{2}(I^{d}) ⊆ L_{2;1}(I^{d}) and X_{kl} = −∂_{k}X_{l}.

Proof It follows from Corollary 2.8 that H_{0}^{−1} is bounded on L_{2}^{⊥}(I^{d}). But if φ ∈ L_{2;1}(I^{d}) then ∂_{k}φ ∈ L_{2}^{⊥}(I^{d}) for each k ∈ {1, …, d}. Hence the x_{l} and x_{kl} are well-defined. Then, however, it follows from [Kat], Theorem VI-3.2, that there is a bounded symmetric operator B on L_{2}^{⊥}(I^{d}) such that

H_{0}^{−1} = (Re H_{0})^{−1/2}(I + iB)^{−1}(Re H_{0})^{−1/2} .  (13)

Therefore

x_{l}(φ) = (φ, (Re H_{0})^{−1/2}(I + iB)^{−1}(Re H_{0})^{−1/2}∂_{l}φ)

and

x_{kl}(φ) = ((Re H_{0})^{−1/2}∂_{k}φ, (I + iB)^{−1}(Re H_{0})^{−1/2}∂_{l}φ) .

Since ∥(I + iB)^{−1}∥_{2→2} ≤ 1 it follows that

|x_{l}(φ)| ≤ ∥(Re H_{0})^{−1/2}∥_{2→2}∥∂_{l}(Re H_{0})^{−1/2}∥_{2→2}∥φ∥_{2}^{2}

and

|x_{kl}(φ)| ≤ ∥∂_{k}(Re H_{0})^{−1/2}∥_{2→2}∥∂_{l}(Re H_{0})^{−1/2}∥_{2→2}∥φ∥_{2}^{2}

where the operator norms are taken on L_{2}^{⊥}(I^{d}). Hence the x_{l} and x_{kl} extend to bounded forms on L_{2}(I^{d}). The associated bounded operators map L_{2;1}(I^{d}) into L_{2}^{⊥}(I^{d}) and since L_{2}^{⊥}(I^{d}) is closed in L_{2}(I^{d}) it implies that the operators map L_{2}(I^{d}) into L_{2}^{⊥}(I^{d}). □

In the sequel we also use the notation X_{l} = X_{l}(C) and X_{kl} = X_{kl}(C) for the operators associated with the forms x_{l} and x_{kl} since it is sometimes necessary to track the dependence on the matrix C of coefficients. Hence X_{l}(C^{∗}) and X_{kl}(C^{∗}) will denote the operators corresponding to the adjoint operator.

The homogenization Ĥ of H is defined to be the elliptic operator on L_{2}(R^{d}) with constant coefficients ĉ_{ij} associated with the form

ĥ(φ) = ∑_{k,l=1}^{d} (∂_{k}φ, ĉ_{kl}∂_{l}φ)

where the coefficients are defined by

ĉ_{kl} = ∫_{I^{d}} du c_{kl}(u) − ∑_{m,n=1}^{d}(\overline{c_{km}}, X_{mn}c_{nl}) .

This definition of Ĥ coincides with that of [BLP], pages 16 and 184, or that of [BBJR]. Note that the homogenized matrix Ĉ = (ĉ_{ij}) satisfies the ellipticity condition (2) with μ = μ_{C} (see, for example, [BLP] Chapter 1, Theorem 3.2). Let K̂ be the kernel of the semigroup Ŝ generated by the operator Ĥ.

Since the coefficients of Ĥ are constant it can be decomposed in a similar manner to H. Then the components Ĥ_{θ}, θ ∈ [−π, π]^{d}, in the decomposition are the maximal accretive operators on L_{2}(I^{d}) associated with the sectorial forms

ĥ_{θ}(φ) = ∑_{k,l=1}^{d}((∂_{k} + iθ_{k})φ, ĉ_{kl}(∂_{l} + iθ_{l})φ)

with D(ĥ_{θ}) = L_{2;1}(I^{d}).

Subsequently, we need the following lemma.

Lemma 2.14 Let φ ∈ L_{2;1}(I^{d}) and τ_{1}, …, τ_{d} ∈ L_{2}(I^{d}). Suppose that

h_{0}(ψ, φ) = ∑_{k=1}^{d}(∂_{k}ψ, τ_{k})

for all ψ ∈ L_{2;1}(I^{d}). Then there is an a ∈ C such that φ = a 11 − ∑_{k=1}^{d} X_{k}τ_{k}.

Proof Since h_{0}(ψ, 11) = 0 we may assume that φ ∈ L_{2}^{⊥}(I^{d}) ∩ L_{2;1}(I^{d}). If χ ∈ L_{2}^{⊥}(I^{d}) then ψ = H_{0}^{∗−1}χ ∈ L_{2;1}(I^{d}) by (13). So

(χ, φ) = h_{0}(ψ, φ) = ∑_{k=1}^{d}(∂_{k}ψ, τ_{k}) = − ∑_{k=1}^{d}(X_{k}^{∗}χ, τ_{k}) = − ∑_{k=1}^{d}(χ, X_{k}τ_{k})

for all χ ∈ L_{2;1}(I^{d}) from which the statement of the lemma follows. □

3 Continuous coefficients

In this section we prove Theorem 1.1. Our derivation of the kernel bounds (3) and (4) is a variation of the argument of Zhikov [Zhi] who considered the asymptotic behaviour of real symmetric operators with periodic measurable coefficients. (Zhikov’s argument is described in Section 2.6 of [ZKO].) In the real symmetric case one has good Gaussian bounds on the kernel and Zhikov used spectral estimates to obtain an asymptotic approximation of the kernel. Throughout this section we assume that the coefficients are continuous. Then the coefficients are uniformly continuous and one has the Gaussian-type bounds of [Aus], Theorem 4.3,

|K_{t}(x; y)| ≤ c (1 + t)^{N}G_{b,t}(x − y)  (14)

for some N ∈ N and b, c > 0 uniformly for all x, y ∈ R^{d} and t > 0. We will combine these with a variation of Zhikov’s spectral estimates to obtain improved bounds.

3.1 Gaussian bounds

In this subsection we prove the first statement of Theorem 1.1. First, note that by the decomposition theory of Section 2 the semigroups S^{θ} are given by bounded integral kernels K^{θ} which are related to K by

K_{t}^{θ}(u; v) = ∑_{n∈Z^{d}} e^{i(n−u+v)·θ}K_{t}(u; v + n) = ∑_{n∈Z^{d}} e^{i(n−u+v)·θ}K_{t}(u − n; v)  (15)

for all u, v ∈ I^{d} and θ ∈ [−π, π]^{d}. (This formula differs from the similar one given in [BJR], Theorem 2.4, by the presence of the factor e^{−i(u−v)·θ} which arises because of the current unitarily equivalent description.) The sums in (15) are convergent for all θ ∈ C^{d} and u, v ∈ I^{d}, so one can use (15) to define a function K_{t}^{θ} for all θ ∈ C^{d}. Then we know that K_{t}^{θ} is the kernel of S_{t}^{θ} if θ ∈ [−π, π]^{d} and we next argue that this extends to all θ ∈ C^{d}.

It immediately follows from (14), (15) and the Jacobi identity, [ScD] Satz I.10.4, that the K^{θ} satisfy bounds similar to those of K: there exist b, c, ω > 0 such that

|K_{t}^{θ}(u; v)| ≤ c (1 + t)^{N}(1 ∧ t)^{−d/2}e^{−b∥u−v∥^{2}t^{−1}}e^{ω(Im θ)^{2}t}  (16)

for all θ ∈ C^{d}, u, v ∈ I^{d} and t > 0, where

∥x∥ = min{|x − n| : n ∈ Z^{d}}

for all x ∈ R^{d}. For all θ ∈ C^{d} and t > 0 let T_{t}^{θ} denote the operator on L_{2}(I^{d}) which has kernel K_{t}^{θ}. If φ, ψ ∈ C_{c}(I^{d}) and t > 0 then it follows again from the Gaussian estimates (14) that θ ↦ (ψ, T_{t}^{θ}φ) is a holomorphic function from C^{d} into C. But it equals the entire function θ ↦ (ψ, S_{t}^{θ}φ) for all θ ∈ [−π, π]^{d}. Hence (ψ, T_{t}^{θ}φ) = (ψ, S_{t}^{θ}φ) for all θ ∈ C. This implies that T_{t}^{θ} = S_{t}^{θ} and therefore K^{θ} is the kernel of S^{θ} for all θ ∈ C.

It now follows from the well known estimates for the norm of the operator of convolution with a Gaussian that there exist c, ω > 0 such that

∥S_{t}^{θ}∥_{p→q} ≤ c (1 + t)^{N}(1 ∧ t)^{−d(1/p−1/q)/2}e^{ω(Im θ)^{2}t}  (17)

uniformly for all 1 ≤ p ≤ q ≤ ∞, t > 0 and θ ∈ C^{d}, where N is as in (14).

Recall that θ ↦ P_{0}(θ) is a holomorphic function from B_{θ_{0}}^{C} into L(L_{2}(I^{d})) since the H_{θ} form a holomorphic family of operators on L_{2}(I^{d}). But the semigroup estimates give the following stronger conclusion.

Lemma 3.1 The map θ ↦ P_{0}(θ) is holomorphic from B_{θ_{0}}^{C} into L(L_{1}(I^{d}), L_{∞}(I^{d})) and from B_{θ_{0}}^{C} into L(L_{p}(I^{d})) for all p ∈ [1, ∞].

Proof First note that P_{0}(θ) = e^{2λ_{0}(θ)}S_{1}^{θ}P_{0}(θ)S_{1}^{θ}. Hence

∥P_{0}(θ)∥_{1→∞} ≤ e^{2 Re λ_{0}(θ)}∥S_{1}^{θ}∥_{1→2}∥P_{0}(θ)∥_{2→2}∥S_{1}^{θ}∥_{2→∞} .

Therefore θ ↦ ∥P_{0}(θ)∥_{1→∞} is locally uniformly bounded.

Secondly, let ψ, φ ∈ L_{1}(I^{d}) ∩ L_{2}(I^{d}) and set φ(θ) = P_{0}(θ)φ. Then θ ↦ φ(θ) is a holomorphic function from B_{θ_{0}}^{C} into L_{2}(I^{d}) and with derivative ∂φ(θ)/∂θ_{j} ∈ L_{2}(I^{d}) for all j. Moreover, for all θ ∈ B_{θ_{0}}^{C} one has

(ψ, φ(θ + η) − φ(θ) − ∑_{j=1}^{d} η_{j} \frac{∂}{∂θ_{j}} φ(θ)) = (2πi)^{−d} ∫_{Γ^{d}} dξ_{1} … dξ_{d} (ψ, φ(ξ)) ·

· ( ∏_{j=1}^{d}(ξ_{j} − θ_{j} − η_{j})^{−1} − ∏_{j=1}^{d}(ξ_{j} − θ_{j})^{−1} − ∑_{k=1}^{d} \frac{η_{k}}{(ξ_{k} − θ_{k})} ∏_{j=1}^{d}(ξ_{j} − θ_{j})^{−1})

for all η ∈ C^{d} with |η| small enough, where Γ is a positively-oriented circle around the origin with radius 2^{−1}(θ_{0} + max_{k∈{1,...,d}} |θ_{j}|). One readily deduces that there is an M > 0 such that

|(ψ, φ(θ + η) − φ(θ) − ∑_{j=1}^{d} η_{j} \frac{∂}{∂θ_{j}} φ(θ))| ≤ M |η|^{2}∥ψ∥_{1} sup_{ξ∈Γ^{d}} ∥φ(ξ)∥_{∞}
≤ M |η|^{2} sup_{ξ∈Γ^{d}} ∥P_{0}(ξ)∥_{1→∞}∥ψ∥_{1}∥φ∥_{1}

uniformly for all ψ, φ ∈ L_{1}(I^{d}) ∩ L_{2}(I^{d}) and all small |η|. Then ∂φ(θ)/∂θ_{j} ∈ L_{∞}(I^{d}) for all j and

∥φ(θ + η) − φ(θ) − ∑_{j=1}^{d} η_{j} \frac{∂}{∂θ_{j}} φ(θ)∥_{∞} ≤ M |η|^{2} sup_{ξ∈Γ^{d}} ∥P_{0}(ξ)∥_{1→∞}∥φ∥_{1}

uniformly for all φ ∈ L_{1}(I^{d})∩L_{2}(I^{d}) and all small |η|. So θ ↦ φ(θ) is holomorphic from B_{θ_{0}}^{C} into L_{∞}(I^{d}) and the holomorphy of the map θ ↦ P_{0}(θ) from B_{θ_{0}}^{C} into L(L_{1}(I^{d}), L_{∞}(I^{d})) follows immediately since L_{1}(I^{d}) ∩ L_{2}(I^{d}) is dense in L_{1}(I^{d}).

Since L_{∞}(I^{d}) ⊂ L_{1}(I^{d}) and the embedding is continuous it follows that the map θ ↦ ∥P_{0}(θ)∥_{∞→∞} is locally uniformly bounded on B_{θ_{0}}^{C}. Then by duality and interpolation the same is valid for the map θ ↦ ∥P_{0}(θ)∥_{p→p} for all p ∈ [1, ∞]. Then the desired conclusion for the holomorphy of the map θ ↦ P_{0}(θ) from B_{θ_{0}}^{C} into L(L_{p}(I^{d})) follows as above. □

After these preliminaries we come to the first important step in the derivation of the Gaussian bounds. We prove that for large t the essential contribution to the kernel comes from the small θ components in the periodic decomposition and these contributions are dominated by the lowest eigenstate. For δ ∈ ⟨0, θ_{0}⟩ define

P_{0}(B_{δ}) = Φ^{−1}((2π)^{−d} ∫_{B_{δ}}^{⊕} dθ P_{0}(θ))Φ .

Then P_{0}(B_{δ}) is a projection on L_{2}(R^{d}).

Lemma 3.2 There exist μ > 0 and for all δ ∈ ⟨0, θ_{0}⟩ a c > 0 such that

∥S_{t}(I − P_{0}(B_{δ}))∥_{1→∞} ≤ c e^{−μt}

uniformly for all t ≥ 1.

Proof Let μ > 0 be as in Proposition 2.12 and δ ∈ ⟨0, θ_{0}]. Then it follows from (8) and Lemma 2.1 that

∥S_{t}(I − P_{0}(B_{δ}))∥_{1→∞} ≤ (2π)^{−d} ∫_{[−π,π]^{d}\B_{δ}} dθ ∥S_{t}^{θ}∥_{1→∞} + (2π)^{−d} ∫_{B_{δ}} dθ ∥S_{t}^{θ}(I − P_{0}(θ))∥_{1→∞}

for all t > 0. We estimate the two terms separately.

It follows from Corollary 2.5, the estimates (17) and the semigroup property that there exists a c > 0 such that ∥S_{t}^{θ}∥_{1→∞} ≤ c e^{−2^{−1}μ_{C}δ^{2}t} uniformly for all t ≥ 1 and θ ∈ [−π, π]^{d}\B_{δ}. So

(2π)^{−d} ∫_{[−π,π]^{d}\B_{δ}} dθ ∥S_{t}^{θ}∥_{1→∞} ≤ c e^{−2^{−1}μ_{C}δ^{2}t}

for all t ≥ 1.

Alternatively, for the integral over B_{δ} we use the semigroup property. Then

∥S_{t}^{θ}(I − P_{0}(θ))∥_{1→∞} ≤ ∥S_{t/3}^{θ}∥_{1→2}∥S_{t/3}^{θ}(I − P_{0}(θ))∥_{2→2}∥S_{t/3}^{θ}∥_{2→∞}

for all θ ∈ [−π, π]^{d} and t > 0. But ∥P_{0}(θ) − P_{0}(0)∥_{2→2} ≤ 1 for all θ ∈ B_{θ_{0}}^{C}. Hence ∥S_{t/3}^{θ}(I − P_{0}(θ))∥_{2→2} ≤ 2e^{−3^{−1}μt} uniformly for all t > 0 and θ ∈ B_{δ}, by Proposition 2.12. Again using the estimates (17) it follows that there is a c′ > 0 such that

∥S_{t}^{θ}(I − P_{0}(θ))∥_{1→∞} ≤ c′ e^{−4^{−1}μt}

uniformly for all t ≥ 1 and θ ∈ B_{δ}. Combination of these estimates completes the proof.□

The second important step in the proof of Gaussian bounds is the comparison of the kernel K with the corresponding kernel K̂ of the homogenized system. Now the semigroup Ŝ generated by the homogenized operator Ĥ can be approximated as in Lemma 3.2 but with the eigenvalue λ̂_{0}(θ) and eigenprojector P̂_{0}(θ) corresponding to Ĥ_{θ}. Hence to compare K and K̂ one must compare the eigenvalues λ_{0}, λ̂_{0} and the eigenprojectors P_{0}, P̂_{0}. The next proposition gives the key eigenvalue comparison.

Let ε̂_{0}, θ̂_{0} and λ̂_{0} be the parameters of Proposition 2.9 corresponding to the operator Ĥ.

Proposition 3.3 There exists a c > 0 such that |λ_{0}(θ) − λ̂_{0}(θ)| ≤ c |θ|^{3} uniformly for θ ∈ B_{2^{−1}(θ_{0}∧θ̂_{0})}^{C}.

Proof Since Ĥ_{θ} is an operator with constant coefficients one has Ĥ_{θ}11 = (θ, Ĉθ)11. So λ̂_{0}(θ) = (θ, Ĉθ) for all θ ∈ BC ; θ̂_{0} by the last statement of Proposition 2.9.

The maps λ_{0} and φ_{0} are holomorphic on B_{θ_{0}}^{C}, so by Lemma 2.10 for all α ∈ J(d) = ⋃_{n=0}^{∞}{1, …, d}^{n} there are λ_{α} ∈ C and φ_{α} ∈ L_{2;1}(I^{d}) such that

λ_{0}(θ) = ∑_{α∈J(d)} λ_{α} θ^{α} and φ_{0}(θ) = ∑_{α∈J(d)} θ^{α} φ_{α}

for all θ ∈ B_{θ_{0}}^{C}. The second series converges both in L_{2}(I^{d}) and in L_{2;1}(I^{d}). Then λ_{α} = 0 and φ_{α} = 11 if |α| = 0, since λ_{0}(0) = 0 and φ_{0}(0) = 11, where |α| = n if α = (i_{1}, …, i_{n}) ∈ J(d). But for all ψ ∈ L_{2;1}(R^{d}) one has

∑_{β∈J(d)} λ_{β} θ^{β} ∑_{γ∈J(d)}(ψ, θ^{γ} φ_{γ}) = λ_{0}(θ) (ψ, φ_{0}(θ)) = (ψ, H_{θ}φ_{0}(θ))

= ∑_{k,l=1}^{d} ∑_{α∈J(d)}((∂_{k} + i\overline{θ_{k}})ψ, c_{kl}(∂_{l} + iθ_{l}) θ^{α} φ_{α})  (18)

for all θ ∈ B_{θ_{0}}^{C}.

Let m ∈ {1, …, d}. Then comparing coefficients of θ_{m} gives

λ_{(m)} (ψ, 11) = h_{0}(ψ, φ_{(m)}) + ∑_{k=1}^{d} i (∂_{k}ψ, c_{km}11) .

Taking ψ = 11 gives λ_{(m)} = 0. Moreover,

h_{0}(ψ, φ_{(m)}) = −i ∑_{k=1}^{d}(∂_{k}ψ, c_{km}11)  (19)

for all ψ ∈ L_{2;1}(I^{d}) and hence

∂_{l}φ_{(m)} = −i ∑_{k=1}^{d} X_{lk}c_{km}  (20)

for all l ∈ {1, …, d}, by Lemmas 2.13 and 2.14.

Next let m, n ∈ {1, …, d} with m ≠ n. Then comparing coefficients of θ_{m}θ_{n} in (18) gives

λ_{(m,n)} (ψ, 11) + λ_{(n,m)} (ψ, 11) = h_{0}(ψ, φ_{(m,n)}) + h_{0}(ψ, φ_{(n,m)})

+ ∑_{l=1}^{d}(iψ, c_{ml}∂_{l}φ_{(n)}) + ∑_{k=1}^{d}(∂_{k}ψ, c_{km}iφ_{(n)})

+ ∑_{l=1}^{d}(iψ, c_{nl}∂_{l}φ_{(m)}) + ∑_{k=1}^{d}(∂_{k}ψ, c_{kn}iφ_{(m)})

+ (ψ, c_{mn}11) + (ψ, c_{nm}11)

for all ψ ∈ L_{2;1}(I^{d}). Substituting ψ = 11 one finds

λ_{(m,n)} + λ_{(n,m)} = −i ∑_{l=1}^{d}(11, c_{ml}∂_{l}φ_{(n)}) − i ∑_{l=1}^{d}(11, c_{nl}∂_{l}φ_{(m)}) + ∫_{I^{d}} c_{mn} + ∫_{I^{d}} c_{nm}

= − ∑_{k,l=1}^{d}(11, c_{ml}X_{lk}c_{kn}) − ∑_{k,l=1}^{d}(11, c_{nl}X_{lk}c_{km}) + ∫_{I^{d}} c_{mn} + ∫_{I^{d}} c_{nm}
= ĉ_{mn} + ĉ_{nm}

by (20). Similarly,

λ_{(n,n)} = − ∑_{k,l=1}^{d}(11, c_{nl}X_{lk}c_{kn}) + ∫_{I^{d}} c_{nn} = ĉ_{nn}

for all n ∈ {1, …, d}. So

λ_{0}(θ) = λ̂_{0}(θ) + ∑_{|α|≥3} λ_{α} θ^{α}

for all θ ∈ B_{θ_{0}∧θ̂_{0}}^{C} and the proposition follows. □

The above calculation also gives the first-order term in the Taylor expansion of the holomorphic function θ ↦ P_{0}(θ). For m ∈ {1, …, d} define

χ_{m} = ∑_{k=1}^{d} X_{k}(C)c_{km} ∈ L_{2}(I^{d}) and χ_{m}^{t} = ∑_{k=1}^{d} X_{k}(C^{t})c_{km}^{t} ∈ L_{2}(I^{d}) ,

where C^{t} = (c_{km}^{t}) with c_{km}^{t} = c_{mk} denotes the transpose matrix. These functions are the first-order correctors of homogenization theory ([BLP], [ZKO]). For θ ∈ B_{θ_{0}}^{C} and m ∈ {1, …, d} let P_{0}^{(m)}(θ) = ∂P_{0}(θ)/∂θ_{m}.

Corollary 3.4 If m ∈ {1, …, d} then χ_{m}, χ_{m}^{t} ∈ L_{∞}(I^{d}). Moreover,

P_{0}^{(m)}(0) φ = i χ_{m} P_{0}(0) φ − i P_{0}(0) (χ_{m}^{t} φ)

for all φ ∈ L_{2}(I^{d}).

Proof If φ_{0}(θ) = P_{0}(θ)11 and φ_{0}^{†}(θ) = P_{0}(θ)^{∗}11 then

(ψ, P_{0}(θ)φ) = \frac{(ψ, φ_{0}(θ))(φ_{0}^{†}(θ), φ)}{(φ_{0}^{†}(θ), φ_{0}(θ))} .

Hence we must calculate the coefficient of θ_{m} on the right hand side and identify it with the expression for (ψ, P_{0}^{(m)}(0)φ). But it follows from (19) and Lemma 2.14 that

φ_{0}(θ) = 11 + i ∑_{m=1}^{d} θ_{m}( ∑_{k=1}^{d} X_{k}(C)c_{km} + a_{m} 11) + O(|θ|^{2})

= 11 + i ∑_{m=1}^{d} θ_{m} 11(χ_{m} + a_{m} 11) + O(|θ|^{2})

for some a ∈ C^{d}. Similarly

φ_{0}^{†}(θ) = 11 + i ∑_{m=1}^{d} \overline{θ_{m}}( ∑_{k=1}^{d} \overline{X_{k}(C^{t})} \overline{c_{km}^{t}} + a_{m}^{†} 11) + O(|θ|^{2})

= 11 + i ∑_{m=1}^{d} \overline{θ_{m}}(\overline{χ_{m}^{t}} + a_{m}^{†} 11) + O(|θ|^{2})

for some a^{†} ∈ C^{d}. Hence

(φ_{0}^{†}(θ), φ_{0}(θ)) = 1 + i ∑_{m=1}^{d}(a_{m} − \overline{a_{m}^{t}})θ_{m} + O(|θ|^{2})

because (11, X_{k}(C)c_{km}) = 0 = (X_{k}(C^{t})c_{km}^{t}, 11). Therefore

(ψ, P_{0}^{(m)}(0)φ) = (ψ, i χ_{m})(11, φ) + (ψ, 11)(i \overline{χ_{m}^{t}}, φ)  (21)

for all φ, ψ ∈ L_{2}(I^{d}), by a simple calculation. The a_{m}, a_{m}^{†} give no contribution to the first order terms.

Next, the map θ ↦ P_{0}(θ) is holomorphic from B_{θ_{0}}^{C} into L(L_{1}(I^{d}), L_{∞}(I^{d})), by Lemma 3.1. So P_{0}^{(m)}(θ) ∈ L(L_{1}(I^{d}), L_{∞}(I^{d})) for all θ ∈ B_{θ_{0}}^{C}. Therefore χ_{m} = −i P_{0}^{(m)}(0)11 ∈ L_{∞}(I^{d}) since χ_{m}^{t} has mean value zero. Replacing C by C^{t} gives the same for χ_{m}^{t}. Note that this implies that

φ_{0}(θ) = 11 + i ∑_{m=1}^{d} θ_{m} χ_{m} + O(|θ|^{2})  (22)

both in the L_{2}-, and the L_{∞}-sense. The last statement rephrases (21). □

The most important step in the proof of Gaussian bounds is to show that the homogenized kernel K̂ gives a first-order asymptotic approximation to K.

Theorem 3.5 There exists a c > 0 such that

∥K_{t} − K̂_{t}∥_{∞} ≤ c t^{−1/2} t^{−d/2}

uniformly for all t ≥ 1.

Proof Since Ĥ_{θ}11 = (θ, Ĉθ)11 if follows that P̂_{0}(θ) is a (possibly non-orthogonal) projection onto the constant functions. But similarly (Ĥ_{θ})^{∗}11 = (θ, Ĉθ)^{∗}11 and hence P̂_{0}(θ)^{∗} is also a projection onto the constant functions. Therefore P̂_{0}(θ) = P_{0}(0) is the orthogonal projection onto the constant functions for all θ ∈ BC ; θ̂_{0}.

Next, by Lemma 3.2 there exist c, μ > 0 such that

∥S_{t}(I − P_{0}(B_{δ}))∥_{1→∞} ≤ c e^{−μt} and ∥Ŝ_{t}(I − P̂_{0}(B_{δ}))∥_{1→∞} ≤ c e^{−μt}

uniformly for all t ≥ 1, where δ = 2^{−1}(θ_{0} ∧ θ̂_{0}). Then

∥S_{t} − Ŝ_{t}∥_{1→∞} ≤ 2c e^{−μt} + ∥S_{t}P_{0}(B_{δ}) − Ŝ_{t}P̂_{0}(B_{δ})∥_{1→∞}

for all t ≥ 1. Hence it follows from Lemma 2.1 that one needs to estimate ∥e^{−λ_{0}(θ)t}P_{0}(θ) − e^{−λ̂_{0}(θ)t}P_{0}(0)∥_{1→∞} for all θ ∈ B_{δ}. Obviously one has the decomposition

∥e^{−λ_{0}(θ)t}P_{0}(θ) − e^{−λ̂_{0}(θ)t}P_{0}(0)∥_{1→∞} ≤ |e^{−λ_{0}(θ)t} − e^{−λ̂_{0}(θ)t}| ∥P_{0}(0)∥_{1→∞}
+ |e^{−λ_{0}(θ)t}| ∥P_{0}(θ) − P_{0}(0)∥_{1→∞} (23)

and we estimate the two terms separately.

The first term is easily handled with the eigenvalue estimates of Proposition 3.3. It follows from this proposition, Corollary 2.6 and a Duhamel estimate that

|e^{−λ_{0}(θ)t} − e^{−λ̂_{0}(θ)t}| = |t ∫_{0}^{1} ds e^{−t(sλ_{0}(θ)+(1−s)λ̂_{0}(θ))}(λ̂_{0}(θ) − λ_{0}(θ))|
≤ c t ∫_{0}^{1} ds e^{−tμ_{C}|θ|^{2}}|θ|^{3} = c t |θ|^{3} e^{−tμ_{C}|θ|^{2}}

for a suitable c > 0 uniformly for all t > 0 and θ ∈ B_{δ}.

Next we consider the second term in (23). It follows from Lemma 3.1 that there is a c > 0 such that

∥P_{0}(θ) − P_{0}(0)∥_{1→∞} ≤ c |θ|

for all θ ∈ B_{δ}. Hence one deduces from Corollary 2.6 that there is a c′ > 0 such that

∥e^{−λ_{0}(θ)t}P_{0}(θ) − e^{−λ̂_{0}(θ)t}P_{0}(0)∥_{1→∞} ≤ c′(|θ| + t |θ|^{3})e^{−μ_{C}|θ|^{2}t}  (24)

uniformly for all t > 0 and θ ∈ B_{δ}. Then

∥S_{t}P_{0}(B_{δ}) − Ŝ_{t}P̂_{0}(B_{δ})∥_{1→∞} ≤ c′ ∫_{B_{δ}} dθ (|θ| + t |θ|^{3})e^{−μ_{C}|θ|^{2}t}
≤ c′ ∫_{R^{d}} dθ (|θ| + t |θ|^{3})e^{−μ_{C}|θ|^{2}t} = c′′ t^{−1/2} t^{−d/2} ,

where c′′ = c′ ∫_{R^{d}} dθ e^{−μ_{C}|θ|^{2}} (|θ| + |θ|^{3}).

It now follows by combination of these estimates that

∥S_{t} − Ŝ_{t}∥_{1→∞} ≤ 2c e^{−μt} + c′′ t^{−1/2} t^{−d/2}  (25)

uniformly for all t ≥ 1. Thus one can now choose a c′ > 0 such that

∥K_{t} − K̂_{t}∥_{∞} = ∥S_{t} − Ŝ_{t}∥_{1→∞} ≤ c′ t^{−1/2} t^{−d/2}

uniformly for all t ≥ 1. □

By interpolation one deduces Gaussian bounds for the difference of K and K̂, together with a multiplicative factor which is almost t^{−1/2}.

Corollary 3.6 For all ε > 0 there exist b, c > 0 such that

|K_{t}(x; y) − K̂_{t}(x − y)| ≤ c t^{−(1−ε)/2} G_{b,t}(x − y)

uniformly for all t ≥ 1 and x, y ∈ R^{d}.

Proof By Theorem 3.5 there exists a c > 0 such that

|K_{t}(x; y) − K̂_{t}(x − y)| ≤ c t^{−1/2} t^{−d/2}

uniformly for all t ≥ 1 and x, y ∈ R^{d}. Next K satisfies the bounds (14) and because Ĥ is a strongly elliptic operator with constant coefficients K̂ satisfies Gaussian bounds

|K̂_{t}(x − y)| ≤ c′ G_{b,t}(x − y) .

Therefore there exist b, c′ > 0 and N ∈ N such that

|K_{t}(x; y) − K̂_{t}(x − y)| ≤ c′ t^{N} t^{−d/2} e^{−b|x−y|^{2}t^{−1}}

uniformly for all t ≥ 1 and x, y ∈ R^{d}. Then

|K_{t}(x; y) − K̂_{t}(x − y)| = |K_{t}(x; y) − K̂_{t}(x − y)|^{ε}|K_{t}(x; y) − K̂_{t}(x − y)|^{1−ε}
≤ (c′)^{ε} c^{1−ε}t^{−(1−ε(1+2N))/2} t^{−d/2} e^{−bε|x−y|^{2}t^{−1}}

for all ε ∈ ⟨0, 1], t ≥ 1 and x, y ∈ R^{d}. The proof of the corollary is complete. □

Finally the Gaussian bounds of Theorem 1.1.I are an immediate consequence. Since K̂ satisfies Gaussian bounds uniformly for t > 0 the bounds follow for large t from Corollary 3.6. But for small t the bounds are well known.

Note that it follows immediately that one can omit the growth factor (1 + t)^{N} in the pointwise bounds (16) for the kernel K^{θ}. Then, however, one has bounds

∥S_{t}^{θ}∥_{2→∞} = sup_{u∈I^{d}} ( ∫_{I^{d}} dv |K_{t}^{θ}(u; v)|^{2})^{1/2} ≤ c (1 ∧ t)^{−d/4}

for all θ ∈ [−π, π]^{d} and t > 0. A duality argument gives similar bounds on ∥S_{t}^{θ}∥_{1→2}. Then it follows from Corollary 2.5 and the semigroup property that

|K_{t}^{θ}(u; v)| ≤ ∥S_{t}^{θ}∥_{1→∞} ≤ c′ (1 ∧ t)^{−d/2}e^{−ω|θ|^{2}t}

for all θ ∈ [−π, π]^{d} with c′, ω > 0. Combination of these estimates with the pointwise Gaussian bounds yields the following conclusion.

Corollary 3.7 There exist b, c, ω > 0 such that

|K_{t}^{θ}(u; v)| ≤ c (1 ∧ t)^{−d/2}e^{−b∥u−v∥^{2}t^{−1}}e^{−ω|θ|^{2}t}  (26)

for all θ ∈ [−π, π]^{d}, u, v ∈ I^{d} and t > 0.

The estimates (26) allow one to deduce from [ADM], Theorem G, and [DuR], Theorem 3.4, that there is an ω > 0 such that H_{θ} −ω|θ|^{2}I has a bounded H_{∞}-functional calculus on L_{p}(I^{d}) for each p ∈ ⟨1, ∞⟩. In fact it follows from the proofs of these results that the estimates in the functional calculus are uniform for θ ∈ [−π, π]^{d}. But for our purposes it suffices to note that the bounds (26) imply that there are c, ω > 0 such that

∥S_{t}^{θ}∥_{p→p} ≤ c e^{−ω|θ|^{2}t}  (27)

uniformly for p ∈ [1, ∞], t > 0 and θ ∈ [−π, π]^{d},

3.2 Hölder estimates

In this subsection we prove the second statement of Theorem 1.1, the Gaussian bounds (4) on the Hölder derivatives of the kernel K. The proof is similar in outline to the proof of Gaussian bounds in Subsection 3.1. We begin with the bounds of Auscher, [Aus] Theorem 4.3, which establish that for all ν ∈ ⟨0, 1⟩ there exist b, c > 0 and N ∈ N such that

|K_{t}(x − h; y) − K_{t}(x; y)| ≤ c (|h|t^{−1/2})^{ν}(1 + t)^{N}G_{b,t}(x − y)  (28)

for all h, x, y ∈ R^{d} and t > 0 with |h| ≤ t^{1/2}. These bounds rely on the uniform continuity of the coefficients. We use the spectral estimates to improve these bounds by removing the growth factor. The tactic is to estimate the difference between the Hölder derivative of K and the Hölder derivative of K̂.

Theorem 3.8 There exists a c > 0 such that

|(K_{t} − K̂_{t})(x − h; y) − (K_{t} − K̂_{t})(x; y)| ≤ c t^{−(1−ν)/2}(|h|t^{−1/2})^{ν} t^{−d/2}

uniformly for all t ≥ 1 and x, y, h, ∈ R^{d}.

Proof The basic idea of the proof is to prove first uniform bounds

∥(I − U(h))(S_{t} − Ŝ_{t})∥_{1→∞} ≤ c |h|^{ν}t^{−1/2} t^{−d/2}  (29)

uniformly for all h ∈ R^{d} and t ≥ 1 with |h| ≤ 1 and then remove the restriction on |h|.

Again we use decomposition theory to isolate the principal contribution which comes from the lowest eigenstate. This is an elaboration of the arguments of Subsection 3.1 but the decomposition of translations introduces an additional term.

Fix ν ∈ ⟨0, 1⟩. If δ = 2^{−1}(θ_{0} ∧ θ̂_{0}), as before, then by Lemma 3.2 there exist c, μ > 0 such that ∥S_{t}(I − P_{0}(B_{δ}))∥_{1→∞} ≤ c e^{−μt} uniformly for all t ≥ 1. Then there is a c′ > 0 such that

∥(I −U(h))S_{t}(I −P_{0}(B_{δ}))∥_{1→∞} ≤ ∥(I −U(h))S_{1}∥_{∞→∞}∥S_{t−1}(I −P_{0}(B_{δ}))∥_{1→∞} ≤ c′ |h|^{ν}e^{−μt}

uniformly for all h ∈ R^{d} with |h| ≤ 1 and t ≥ 2. Since similar estimates are valid for Ŝ we will have proved (29) once we can show that there is a c > 0 such that

∥(I − U(h))(S_{t}P_{0}(B_{δ}) − Ŝ_{t}P̂_{0}(B_{δ}))∥_{1→∞} ≤ c |h|^{ν}t^{−1/2} t^{−d/2}

uniformly for all h ∈ R^{d} and t ≥ 1 with |h| ≤ 1.

If φ ∈ L_{2}(I^{d}) and h ∈ R^{d} define L(h)φ ∈ L_{2}(I^{d}) by (L(h)φ)(u) = τ(e^{2πi(u−h)}) where τ ∈ L_{2}(T^{d}) is given by τ(e^{2πiu}) = φ(u). Then L(h) is a unitary operator and it follows that the operator U of translations on R^{d} has the decomposition

Φ U(h) Φ^{−1} = (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ e^{−iθ·h}L(h)  (30)

for all h ∈ R^{d}. Therefore

Φ(I − U(h))(S_{t}P_{0}(B_{δ}) − Ŝ_{t}P̂_{0}(B_{δ}))Φ^{−1}
= (2π)^{−d} ∫_{B_{δ}} dθ (1 − e^{−iθ·h})(e^{−λ_{0}(θ)t}P_{0}(θ) − e^{−λ̂_{0}(θ)t}P_{0}(0))
+ (2π)^{−d} ∫_{B_{δ}} dθ e^{−iθ·h}(I − L(h))(e^{−λ_{0}(θ)t}P_{0}(θ) − e^{−λ̂_{0}(θ)t}P_{0}(0))

for all h ∈ R^{d} and t > 0. Then by Lemma 2.1

∥(I − U(h))(S_{t}P_{0}(B_{δ}) − Ŝ_{t}P̂_{0}(B_{δ}))∥_{1→∞}
≤ (2π)^{−d} ∫_{B_{δ}} dθ |1 − e^{−iθ·h}| ∥e^{−λ_{0}(θ)t}P_{0}(θ) − e^{−λ̂_{0}(θ)t}P_{0}(0)∥_{1→∞}
+ (2π)^{−d} ∫_{B_{δ}} dθ ∥(I − L(h))(e^{−λ_{0}(θ)t}P_{0}(θ) − e^{−λ̂_{0}(θ)t}P_{0}(0))∥_{1→∞} . (31)

The first term on the right hand side is the additional contribution from the decomposition of the translation. If c′ is as in (24) then

(2π)^{−d} ∫_{B_{δ}} dθ |1 − e^{−iθ·h}| ∥e^{−λ_{0}(θ)t}P_{0}(θ) − e^{−λ̂_{0}(θ)t}P_{0}(0)∥_{1→∞}
≤ (2π)^{−d}c′ ∫_{B_{δ}} dθ |h| |θ| (|θ| + t |θ|^{3})e^{−μ_{C}|θ|^{2}t}
≤ c′ |h| ∫_{R^{d}} dθ |θ| (|θ| + t |θ|^{3})e^{−μ_{C}|θ|^{2}t}
= c′′ |h| t^{−(d+2)/2} ≤ c′′t^{−1/2} (|h|t^{−1/2})^{ν} t^{−d/2}

for all h ∈ R^{d} and t > 0 with |h| ≤ t^{1/2} and c′′ = c′ ∫_{R^{d}} dθ e^{−μ_{C}|θ|^{2}} |θ|(|θ| + |θ|^{3}).

The second term on the right hand side of (31) can be estimated by the reasoning of Subsection 3.1. First we need some preliminary estimates comparable to (17).

If one extends the function u ↦ K_{t}^{θ}(u; v) to a periodic function on R^{d} for all v ∈ I^{d} then

K_{t}^{θ}(x; v) = ∑_{n∈Z^{d}} e^{i(n−x+v)·θ}K_{t}(x − n; v)

for all θ ∈ C^{d}, t > 0, x ∈ R^{d} and v ∈ I^{d}. Using the bounds (28) and the Gaussian bounds of Theorem 1.1.I it follows that there are b, c, ω > 0 and N ∈ N such that

|K_{t}^{θ}(u − h; v) − K_{t}^{θ}(u; v)| ≤ ∑_{n∈Z^{d}} |e^{i(n−u+h+v)·θ} − e^{i(n−u+v)·θ}| |K_{t}(u − n; v)|

+ ∑_{n∈Z^{d}} |e^{i(n−u+h+v)·θ}| |K_{t}(u − h − n; v) − K_{t}(u − n; v)|

≤ c |h| t^{1/2}(1 ∧ t)^{−d/2}e^{−b∥u−v∥^{2}t^{−1}}e^{ω(Im θ)^{2}t}

+ c (|h|t^{−1/2})^{ν}(1 + t)^{N}(1 ∧ t)^{−d/2}e^{−b∥u−h−v∥^{2}t^{−1}}e^{ω(Im θ)^{2}t}

uniformly for all θ ∈ C^{d}, t > 0, h ∈ R^{d} and u, v ∈ I^{d} with |h| ≤ t^{1/2} ∧ 1. Since (u, v) ↦ K^{θ}(u−h; v) is the kernel of the operator L(h) S_{t}^{θ} it follows that there are c, ω > 0 such that

∥(I − L(h))S_{t}^{θ}∥_{p→q} ≤ c (|h|t^{−1/2})^{ν}(1 + t)^{N}(1 ∧ t)^{−d(1/p−1/q)/2}e^{ω(Im θ)^{2}t}  (32)

uniformly for all 1 ≤ p ≤ q ≤ ∞, θ ∈ C^{d}, t > 0 and h ∈ R^{d} with |h| ≤ t^{1/2} ∧ 1. Next define the Banach space

C^{ν} = {φ ∈ C(I^{d}) : sup_{0<|h|≤1} |h|^{−ν} ∥(I − L(h))φ∥_{∞} < ∞}

with norm ∥φ∥_{C^{ν}} = ∥φ∥_{∞}+sup_{0<|h|≤1} |h|^{−ν} ∥(I−L(h))φ∥_{∞}. Note that φ(u_{1}, …, 0, …, u_{d}) = φ(u_{1}, …, 1, …, u_{d}) for all φ ∈ C^{ν}, u ∈ I^{d} and k ∈ {1, …, d}, where the 0 and 1 are in the k-th position. Then (32) together with (17) implies that S_{t}^{θ} is a bounded operator from L_{1}(I^{d}) into C^{ν} and

∥S_{t}^{θ}∥_{L_{1}(I^{d})→C^{ν}} ≤ c (1 + t)^{N}(1 ∧ t)^{−d(1/p−1/q)/2}e^{ω(Im θ)^{2}t}

uniformly for all 1 ≤ p ≤ q ≤ ∞, θ ∈ C^{d} and t > 0. In particular, the family of operators is bounded locally uniform in θ. Arguing as in the proof of Lemma 3.1 it follows that the spectral projections θ ↦ P_{0}(θ) form a holomorphic family of operators from B_{θ_{0}}^{C} into L(L_{1}(I^{d}), C^{ν}). Hence there is a c > 0 such that ∥P_{0}(θ) − P_{0}(0)∥_{L_{1}(I^{d})→C^{ν}} ≤ c |θ| uniformly for all θ ∈ B_{δ}. In particular

∥(I − L(h))(P_{0}(θ) − P_{0}(0))∥_{1→∞} ≤ c |θ| |h|^{ν}

for all θ ∈ B_{δ} and h ∈ R^{d} with |h| ≤ 1. This gives estimates, as in the proof of Theorem 3.5,

(2π)^{−d} ∫_{B_{δ}} dθ ∥(I − L(h))(e^{−λ_{0}(θ)t}P_{0}(θ) − e^{−λ̂_{0}(θ)t}P_{0}(0))∥_{1→∞} ≤ c′ t^{−1/2} |h|^{ν} t^{−d/2}

uniformly for all h ∈ R^{d} and t ≥ 1 with |h| ≤ 1 which bound the second term in (31). Adding the contributions it follows that there exists a c > 0 such that

∥(I − U(h))(S_{t} − Ŝ_{t})∥_{1→∞} ≤ c t^{−1/2} |h|^{ν} t^{−d/2}  (33)

uniformly for all h ∈ R^{d} and t ≥ 2 with |h| ≤ 1. But then it follows from (25) that (33) is also valid for all h ∈ R^{d} with |h| ≥ 1 and by (28) also for t ∈ [1, 2], with a possibly larger value of c. So the estimates (33) are valid uniformly for all h ∈ R^{d} and t ≥ 1 and the theorem follows. □

With several applications of interpolation one can deduce rather sharp Gaussian bounds for the Hölder derivatives of the difference.

Theorem 3.9 For all ε, ν, τ ∈ ⟨0, 1⟩ and κ > 0 there exist b, c > 0 such that

|(K_{t}−K̂_{t})(x−h; y−k)−(K_{t}−K̂_{t})(x; y)| ≤ c t^{−(1−ε)(1−ν)/2}(\frac{|h| + |k|}{|x − y| + t^{1/2}} )^{ν}G_{b,t}(x−y) (34)

uniformly for all t ≥ 1 and x, y, h, k ∈ R^{d} with |h| + |k| ≤ τ|x − y| + κt^{1/2}.

Proof Interpolation of the bounds of Theorem 3.8 with the bounds (28) of Auscher, which are of course also valid for K̂, implies that for all ε ∈ ⟨0, 1⟩ there exist b, c > 0 such that

|(K_{t} − K̂_{t})(x − h; y) − (K_{t} − K̂_{t})(x; y)| ≤ c (|h|t^{−1/2})^{ν} t^{−(1−ε)(1−ν)/2} G_{b,t}(x − y)

uniformly for all t ≥ 1 and h, x, y ∈ R^{d} with |h| ≤ t^{1/2}.

These bounds are a special case of (34). To obtain the general bounds we use two additional arguments.

First it follows from the last bounds that for all ε ∈ ⟨0, 1⟩ there exists a c > 0 such that

∥(I − U(h))(S_{t} − Ŝ_{t})∥_{2→∞} ≤ c (|h|t^{−1/2})^{ν} t^{−(1−ε)(1−ν)/2} t^{−d/4}  (35)

uniformly for all h ∈ R^{d} and t ≥ 1 with |h| ≤ t^{1/2}. But by Theorem 3.5 and increasing the value of c it follows that the bounds (35) are valid uniformly for all h ∈ R^{d} and t ≥ 1. The semigroup Ŝ satisfies similar bounds: there exists a c > 0 such that

∥(I − U(h))Ŝ_{t}∥_{2→∞} ≤ c (|h|t^{−1/2})^{ν} t^{−d/4}

for all h ∈ R^{d} and t > 0. So there is a c > 0 such that

∥(I − U(h))S_{t}∥_{2→∞} ≤ c (|h|t^{−1/2})^{ν} t^{−d/4}

for all h ∈ R^{d} and t ≥ 1. Then by the previous estimates and duality one deduces that for all ε ∈ ⟨0, 1⟩ there exists a c > 0 such that

∥(I − U(h))(S_{t} − Ŝ_{t})(I − U(k))∥_{1→∞} ≤ ∥(I − U(h))S_{t/2}∥_{2→∞}∥(S_{t/2} − Ŝ_{t/2})(I − U(k))∥_{1→2}

+ ∥(I − U(h))(S_{t/2} − Ŝ_{t/2})∥_{2→∞}∥Ŝ_{t/2}(I − U(k))∥_{1→∞}

≤ c (|h| |k| t^{−1})^{ν} t^{−(1−ε)(1−ν)/2} t^{−d/2}

for all h, k ∈ R^{d} and t ≥ 2. Together with the estimates of Theorem 3.8 and duality one establishes that for all ε ∈ ⟨0, 1⟩ there exists a c > 0 such that

|(K_{t} − K̂_{t})(x − h; y − k) − (K_{t} − K̂_{t})(x; y)|

≤ c((|h| |k| t^{−1})^{ν} + (|h|t^{−1/2})^{ν} + (|k|t^{−1/2})^{ν})t^{−(1−ε)(1−ν)/2} t^{−d/2}

uniformly for all h, k, x, y ∈ R^{d} and t ≥ 1.

Finally we can apply Lemma 4.2 of [ElR2]. This lemma allows one to interpolate between the last bounds and the bounds of Corollary 3.6 to obtain the bounds (34) for any slightly smaller ν. □

Obviously Theorem 1.1.II is an immediate consequence of Theorem 3.9.

3.3 Asymptotic estimates

In the foregoing derivation of kernel bounds the estimate of Theorem 3.5 played a key role. The proof of this estimate can, in principle, be extended to obtain a complete asymptotic expansion of K in terms of K̂, the derivatives of K̂ and the correctors of homogenization theory. In this subsection we discuss the first-order term in this expansion.

The zero-order term in the expansion was derived by first making a spectral decomposition, secondly approximating the components in terms of an integral over small |θ| of the contribution coming from the eigenprojector P_{0}(θ) and the eigenvalue λ_{0}(θ) and thirdly approximating the eigenprojector and eigenvalue by the lowest order terms in their holomorphic expansions. The first-order term in the expansion is derived by the same process but arises from the next terms in the holomorphic expansions. In the case of the eigenprojector this is the linear term in θ and for the eigenvalue the term of order θ^{3}. The expansion for the eigenvalue has the form

λ_{0}(θ) = λ̂_{0}(θ) + ∑_{α∈J(d); |α|≥3} λ_{α} θ^{α}

for θ ∈ B_{θ_{0}∧θ̂_{0}}^{C} as a consequence of Proposition 3.3.

Lemma 3.10 For all m ∈ {1, …, d} the correctors χ_{m} and χ_{m}^{t} extend to continuous periodic functions on R^{d}.

Proof It follows from (22) that P_{0}(θ)11 = φ_{0}(θ) = 11 + i ∑_{m=1}^{d} θ_{m} χ_{m} + O(|θ|^{2}) in the L_{∞}-sense. But the map θ ↦ P_{0}(θ) is holomorphic from B_{θ_{0}}^{C} into L(L_{1}(I^{d}), C^{ν}), by the proof of Theorem 3.8. Hence χ_{m} ∈ C^{ν} for all m ∈ {1, …, d} and it extends to a continuous periodic function on R^{d}. □

Proposition 3.11 There exists a c > 0 such that

|K_{t}(x; y) − K̂_{t}(x − y) − ∑_{m=1}^{d} (χ_{m}(x) − χ_{m}^{t}(y))(∂_{m}K̂_{t})(x − y)

− it ∑_{α∈J(d); |α|=3} λ_{α}(∂^{α}K̂_{t})(x − y)| ≤ c t^{−1} t^{−d/2}

uniformly for all t ≥ 1 and x, y ∈ R^{d}.

Remark Each term occurring in the difference estimated in the proposition satisfies a Gaussian bound. Therefore one can interpolate between the uniform bound of the proposition and the Gaussian bounds to obtain a Gaussian bound for the difference. Then, for each ε > 0, one obtains a bound c t^{−(1−ε)}G_{b,t}(x − y) for t ∈ [1, ∞⟩.

Proof of Proposition 3.11 The proof is based on the same reasoning used to prove Theorem 3.5 and we only sketch the main ideas. First it follows from the previous proof that one has bounds

∥S_{t} − Ŝ_{t} − S_{t}P_{0}(B_{δ}) + Ŝ_{t}P̂_{0}(B_{δ})∥_{1→∞} ≤ c e^{−μt}  (36)

for all t ≥ 1. To estimate S_{t}P_{0}(B_{δ}) − Ŝ_{t}P̂_{0}(B_{δ}) we now use the identity

e^{−λ_{0}(θ)t}P_{0}(θ) − e^{−λ̂_{0}(θ))t}P_{0}(0) = e^{−λ̂_{0}(θ)t}((P_{0}(θ) − P_{0}(0)) + t(λ̂_{0}(θ) − λ_{0}(θ))P_{0}(0))
+ te^{−λ̂_{0}(θ)t}(λ̂_{0}(θ) − λ_{0}(θ))(P_{0}(θ) − P_{0}(0))

+ t^{2}(λ̂_{0}(θ) − λ_{0}(θ))^{2} ∫_{0}^{1} ds (1 − s) e^{−λ̂_{0}(θ)t(1−s)−λ_{0}(θ)ts}P_{0}(θ)

which is readily verified with two applications of the Duhamel identity.

The most significant contribution to the operator crossnorm comes from the term

L_{t}(θ) = e^{−λ̂_{0}(θ)t}( ∑_{m=1}^{d} θ_{m}P_{0}^{(m)}(0) − t ∑_{α; |α|=3} λ_{α}θ^{α}P_{0}(0))

arising from the expansion of the bracketed factor in the first term, where P^{(m)} is as in Corollary 3.4 and the λ_{α} as in the proof of Proposition 3.3. All other contributions are bounded by a multiple of t^{−1} t^{−d/2}. But it follows from Corollary 3.4 and the observation that P_{0}(0) = P̂_{0}(θ) for |θ| < δ, that

L_{t}(θ) = ∑_{m=1}^{d} i θ_{m}e^{−λ̂_{0}(θ)t}(χ_{m}P̂_{0}(θ) − P̂_{0}(θ)M_{χ_{m}^{t}}) − i t ∑_{α; |α|=3} λ_{α} (iθ)^{α}e^{−λ̂_{0}(θ)t}P̂_{0}(θ)

where M_{χ_{m}^{t}} denotes the multiplication operator with the function χ_{m}^{t} on the space L_{2}(I^{d}).

If one differentiates (30) then

Φ∂_{m}Ŝ_{t}Φ^{−1} = (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ (iθ_{m} Ŝ_{t}^{θ} + ∂_{m}Ŝ_{t}^{θ})

and

Φ∂_{m}Ŝ_{t}P̂_{0}(B_{δ})Φ^{−1} = (2π)^{−d} ∫_{B_{δ}}^{⊕} dθ ie^{−λ̂_{0}(θ)t}θ_{m} P_{0}(0)

since ∂_{m}P̂_{0}(θ) = ∂_{m}P_{0}(0) = 0. We next determine a decomposition for χ_{m} ∂_{m}Ŝ_{t}P̂_{0}(B_{δ}) for all m ∈ {1, …, d}.

If φ ∈ C_{c}(R^{d}) with Φ(φ) = (φ_{θ})_{θ∈[−π,π]^{d}} then (Φ(φ χ_{m}))(θ) = χ_{m} φ_{θ} for all θ ∈ [−π, π]^{d}. So

Φχ_{m} ∂_{m}Ŝ_{t}P̂_{0}(B_{δ})Φ^{−1} = (2π)^{−d} ∫_{B_{δ}}^{⊕} dθ e^{−λ̂_{0}(θ)t}i θ_{m} χ_{m} P_{0}(0)  (37)

for all t > 0. Similarly the terms with χ_{m}^{t} and third order terms in θ can be handled. Therefore (36) gives an estimate

∥S_{t} − Ŝ_{t} − ∑_{m=1}^{d} (χ_{m}∂_{m}Ŝ_{t} − ∂_{m}Ŝ_{t}M_{χ_{m}^{t}})P̂_{0}(B_{δ}) − it ∑_{α; |α|=3} λ_{α}∂^{α}Ŝ_{t}P̂_{0}(B_{δ})∥_{1→∞} ≤ c t^{−1} t^{−d/2}

uniformly for t ≥ 1. But then P̂_{0}(B_{δ}) can be replaced by the identity because this only introduces exponential corrections. The resulting bounds translate directly into the uniform bounds on the kernels given in the proposition. □

Under some quite general circumstances the first-order correction in the asymptotic expansion of K are zero. Define divC = 0 to mean that ∑_{k=1}^{d} ∂_{k}c_{km} = 0, in the sense of distributions, for each m ∈ {1, …, d}.

Lemma 3.12

I. The correctors χ_{m} = 0 for all m ∈ {1, …, d} if, and only if, divC = 0.

II. The correctors χ_{m}^{t} = 0 for all m ∈ {1, …, d} if, and only if, divC^{t} = 0.

III. If divC = divC^{t} = 0 then λ_{α} = 0 for all α ∈ J(d) with |α| = 3.

Proof First

(H_{0}^{∗}χ, χ_{m}) = − ∑_{k=1}^{d}(∂_{k}χ, c_{km})

for all χ ∈ D(H_{0}^{∗}) ∩ L_{2}^{⊥}(I^{d}) and hence for all χ ∈ D(H_{0}^{∗}) and m ∈ {1, …, d}. But D(H_{0}^{∗}) is dense in L_{2;1}(I^{d}). Hence if χ_{m} = 0 for all m ∈ {1, …, d} then divC = 0. Conversely, if divC = 0 then (H_{0}^{∗}χ, χ_{m}) = 0 for all χ ∈ D(H_{0}^{∗}) since C_{c}^{∞}(I^{d}) is dense in L_{2;1}(I^{d}) and L_{2;1}(I^{d}) ⊆ D(H_{0}^{∗}). As H_{0}^{∗}(L_{2}^{⊥}(I^{d}) ∩ D(H_{0}^{∗})) = L_{2}^{⊥}(I^{d}) and χ_{m} ∈ L_{2}^{⊥}(I^{d}) by Lemma 2.13 it follows that χ_{m} = 0.

The second statement follows from the first by replacing C with C^{t}.

Next for θ ∈ B_{θ_{0}} set R(θ) = λ_{0}(θ) − (θ, Ĉθ). If divC^{t} = 0 then

λ_{0}(θ) (11, P_{0}(θ)11) = (11, H_{θ}P_{0}(θ)11) = (11, (θ, Cθ) P_{0}(θ)11)

and hence

R(θ) = (11, ((θ, Cθ) − (θ, Ĉθ))P_{0}(θ)11)/(11, P_{0}(θ)11) .

Moreover, divC^{t} = 0 implies that Ĉ is the mean of C over I^{d} and hence (11, ((θ, Cθ) − (θ, Ĉθ))P_{0}(0)11) = 0. Therefore

R(θ) = (11, ((θ, Cθ) − (θ, Ĉθ))(P_{0}(θ) − P_{0}(0))11)/(11, P_{0}(θ)11) .

But if, in addition, divC = 0 then ∥P_{0}(θ) − P_{0}(0)∥_{2→2} = O(|θ|^{2}) by Corollary 3.4. Hence |R(θ)| = O(|θ|^{4}). □

Note that if divC^{t} = 0 then the coefficients λ_{(k,l,m)} of the third-order terms in R(θ) are given by

λ_{(k,l,m)} = i ∑_{n=1}^{d}(11, (c_{kl} − (11, c_{kl}11))X_{n} c_{nm})  (38)

where we have use the explicit form of the correctors given in Subsection 2.3 and Corollary 3.4.

The last two results allow one to conclude that the asymptotic estimate (25) is usually optimal.

Proposition 3.13 The following conditions are equivalent.

I. There is a c > 0 such that ∥S_{t} − Ŝ_{t}∥_{1→∞} ≤ c t^{−1} t^{−d/2} uniformly for t ≥ 1.

II. There are c, ε > 0 such that ∥S_{t} − Ŝ_{t}∥_{1→∞} ≤ c t^{−(1+ε)/2} t^{−d/2} uniformly for t ≥ 1.

III. divC = divC^{t} = 0.

Proof It follows from Lemma 3.12 and Proposition 3.11 that III⇒I. But it is evident that I⇒II.

If I is valid then it follows from Proposition 3.11 that there is a c′ > 0 such that

| ∑_{m=1}^{d} (χ_{m}(x) − χ_{m}^{t}(y))(∂_{m}K̂_{t})(x − y) + it ∑_{α∈J(d); |α|=3} λ_{α}(∂^{α}K̂_{t})(x − y)| ≤ c′ t^{−ε/2} t^{−(d+1)/2}

for all t ≥ 1 and x, y ∈ R^{d}. Then

| ∑_{m=1}^{d} (χ_{m}(x) − χ_{m}^{t}(x − t^{1/2}y))(∂_{m}K̂_{1})(y) + i ∑_{α∈J(d); |α|=3} λ_{α}(∂^{α}K̂_{1})(y)|

= t^{(d+1)/2}| ∑_{m=1}^{d} (χ_{m}(x) − χ_{m}^{t}(x − t^{1/2}y))(∂_{m}K̂_{t})(t^{1/2}y) + i t ∑_{α∈J(d); |α|=3} λ_{α}(∂^{α}K̂_{t})(t^{1/2}y)|
≤ c′ t^{−ε/2}

for all x, y ∈ R^{d} and t ≥ 1. Since the χ_{m} and χ_{m}^{t} are periodic functions with mean value zero, integration with respect to x over I^{d} gives

| ∑_{α∈J(d); |α|=3} λ_{α}(∂^{α}K̂_{1})(y)| ≤ c′t^{−ε/2}

for all y ∈ R^{d} and t ≥ 1. Then ∑_{α∈J(d); |α|=3} λ_{α}(∂^{α}K̂_{1})(y) = 0 by taking the limit t → ∞. So

| ∑_{m=1}^{d} (χ_{m}(x) − χ_{m}^{t}(x − t^{1/2}y))(∂_{m}K̂_{1})(y)| ≤ c′t^{−ε/2}

for all x, y ∈ R^{d} and t ≥ 1. Now let x, y_{0} ∈ R^{d}. Since χ_{m}^{t} has again mean value zero it follows by integration with respect to y over the set y_{0} + t^{−1/2}I^{d} that

| ∑_{m=1}^{d} χ_{m}(x) t^{d/2} ∫_{y_{0}+t^{−1/2}I^{d}} dy (∂_{m}K̂_{1})(y)

− ∑_{m=1}^{d} t^{d/2} ∫_{y_{0}+t^{−1/2}I^{d}} dy χ_{m}^{t}(x − t^{1/2}y)((∂_{m}K̂_{1})(y) − (∂_{m}K̂_{1})(y_{0}))|

≤ c′t^{−ε/2}

for all t > 0. But lim_{t→∞} t^{d/2} ∫_{y_{0}+t^{−1/2}I^{d}} dy (∂_{m}K̂_{1})(y) = (∂_{m}K̂_{1})(y_{0}) and |(∂_{m}K̂_{1})(y) − (∂_{m}K̂_{1})(y_{0})| ≤ ∥∇∂_{m}K̂_{1}∥_{∞} |y − y_{0}| = d^{1/2}∥∇∂_{m}K̂_{1}∥_{∞} t^{−1/2} for all y ∈ y_{0} + t^{−1/2}I^{d}. So taking the limit t → ∞ one deduces that

∑_{m=1}^{d} χ_{m}(x) (∂_{m}K̂_{1})(y_{0}) = 0

for all x, y_{0} ∈ R^{d}. Scaling then gives ∑_{m=1}^{d} χ_{m}(x) (∂_{m}K̂_{t})(y) = 0 for all x, y ∈ R^{d} and t > 0. Then for all η, x ∈ R^{d} one has

∑_{m=1}^{d} χ_{m}(x) η_{m} = lim_{t↓0} ∫_{R^{d}} dy ∑_{m=1}^{d} χ_{m}(x) K̂_{t}(y) \frac{∂}{∂y_{m}} e^{η·y}

= − lim_{t↓0} ∫_{R^{d}} dy e^{η·y} ∑_{m=1}^{d} χ_{m}(x) (∂_{m}K̂_{t})(y) = 0

and χ_{m} = 0 for all m ∈ {1, …, d}.

Similarly χ_{m}^{t} = 0 for all m ∈ {1, …, d}. Then Statement I is a consequence of Lemma 3.12. □

Finally we note that the third order terms in the holomorphic expansion of θ ↦ λ_{0}(θ) vanish in the case of symmetric coefficients. We temporarily adopt the notation H_{θ}(C) = H_{θ}, λ_{0}(θ; C), etc. to denote the dependence of the various elements on the matrix of coefficients.

Lemma 3.14 There is a δ > 0 such that

λ_{0}(θ; C) = λ_{0}(−θ; C^{t})

for all θ ∈ B_{δ}^{C}. Hence if C is symmetric, i.e., if C = C^{t}, then θ ↦ λ_{0}(θ; C) is an even function for small |θ|.

Proof Since λ_{0}(θ; C) is an eigenvalue of H_{θ}(C) it follows that \overline{λ_{0}(θ; C)} is an eigenvalue of H_{θ}(C)^{∗}. Let φ_{0}^{∗}(θ; C) be a corresponding eigenfunction. Then

\overline{H_{θ}(C)^{∗}} \overline{φ_{0}^{∗}(θ; C)} = λ_{0}(θ; C) \overline{φ_{0}^{∗}(θ; C)} .

Since \overline{H_{θ}(C)^{∗}} = H_{−θ}(C^{t}) one deduces that

H_{θ}(C) \overline{φ_{0}^{∗}(−θ; C^{t})} = λ_{0}(−θ; C^{t}) \overline{φ_{0}^{∗}(−θ; C^{t})} .

Thus λ_{0}(−θ; C^{t}) is an eigenvalue of H_{θ}(C) such that λ_{0}(−θ; C^{t}) → 0 as |θ| → 0. Hence, by simplicity of the eigenvalue, λ_{0}(−θ; C^{t}) = λ_{0}(θ; C) for all small |θ| (see Proposition 2.9).□

Despite this last result there are non-symmetric examples for which the third-order terms are non-zero. If C = I + εB then H is a perturbation of the Laplacian for small ε. Now for d = 2 set

B(x) = ( \begin{matrix}e^{2πix_{2}} & 0 \\ e^{−2πix_{2}} & 0 \end{matrix} ) .

Then divB^{t} = 0, (11, B11) = 0 and divB(x) = (−2πie^{−2πix_{2}}, 0). Since H_{0}τ = 4π^{2} τ if τ(x) = e^{−2πix_{2}} it follows from (38) that the coefficient λ_{(1,1,1)} = (2π)^{−1}ε^{2} ≠ 0.

3.4 Riesz transforms

In this subsection we establish the last statement of Theorem 1.1, the boundedness of the Riesz transforms. We begin by stating a weaker property, local boundedness, which does not explicitly require periodicity. The following result is essentially contained in Auscher and Tchamitchian, [AuT].

Proposition 3.15 Assume the coefficients c_{kl} of H are bounded, uniformly continuous and that the corresponding semigroup kernel K satisfies Conditions I and II of Theorem 1.1, for some ν ∈ ⟨0, 1⟩. If p ∈ ⟨1, ∞⟩ then D((I + H)^{1/2}) = L_{p;1}(R^{d}) and there exist c_{p}, c′_{p} > 0 such that

c_{p} max_{0≤k≤d} ∥∂_{k}φ∥_{p} ≤ ∥(I + H)^{1/2}φ∥_{p} ≤ c′_{p} max_{0≤k≤d} ∥∂_{k}φ∥_{p}

for all φ ∈ L_{p;1}(R^{d}) where ∂_{0} = I.

Proof First, it follows from [AuT], Proposition III.13, Example III.3.1.1 and Theorem III.3 that H and H^{∗} satisfy the Kato property (K)_{loc}. Secondly, H has the property (G) by assumption. Thirdly, the operator ∂_{k}(I + H)^{−1}∂_{l} extends to a bounded operator on L_{p}(R^{d}) for all k, l ∈ {1, …, d} by Theorem A of [AnMT]. Then the proposition follows from [AuT] Theorem IV.26. □

If H satisfies the conditions of Proposition 3.15 set R_{k}(ε) = ∂_{k}(εI +H)^{−1/2} for ε ∈ ⟨0, 1] and k ∈ {1, …, d}. Then ∥R_{k}(1)∥_{p→p} is finite for each p ∈ ⟨1, ∞⟩ by Proposition 3.15. Moreover, by H_{∞}-functional calculus, one deduces that ∥R_{k}(ε)∥_{p→p} is finite for each ε ∈ ⟨0, 1]. But, in general, the value of the norm diverges as ε → 0. (See the example of Kenig, [AuT] Section IV.2.2, Theorem IV.7.) Next we show that this does not happen if the coefficients are periodic. The proof uses decomposition theory and analysis of the components of the R_{k}(ε). The argument again relies on separation of the contributions from the small θ-components and the lowest eigenvalue of these components.

Step 1 We begin by showing that D((λI + H_{θ})^{1/2}) ⊆ L_{p;1}(I^{d}) and establish a uniform bound for ∥(∂_{k} + iθ_{k})(λI + H_{θ})^{−1/2}∥_{p→p} which is optimal for large λ.

Extend φ ∈ L_{p}(I^{d}) to a function φ̃ over R^{d} by setting φ̃(u−n) = φ(u) for all u ∈ I^{d} and n ∈ Z^{d}. Next let χ_{n} ∈ C_{c}^{∞}(R^{d}) be a sequence of functions with the following properties; first supp χ_{n} ⊆ {x : −n − 1 < x_{k} ≤ n + 1 for all k ∈ {1, …, d}}, secondly 0 ≤ χ_{n} ≤ 1, thirdly χ_{n}(x) = 1 for all x ∈ [−n, n]^{d} and fourthly sup_{n,k} ∥∂_{k}χ_{n}∥_{∞} < ∞. Then

(ψ, φ) = lim_{n→∞}(2n)^{−d}(χ_{n}ψ̃, χ_{n}φ̃)

and

lim_{n→∞}(2n)^{−d}((∂_{k}χ_{n})ψ̃, χ_{n}φ̃) = 0

for all ψ ∈ L_{q}(I^{d}) and φ ∈ L_{p}(R^{d}). Moreover, if φ ∈ ⋂_{k=1}^{d} D(∂_{k}) = L_{p;1}(I^{d}) then χ_{n}φ̃ ∈ ⋂_{k=1}^{d} D(∂_{k}) = L_{p;1}(R^{d}). One verifies immediately that

(H^{(p)∗}τ, χ_{n}φ̃) − (τ, χ_{n}(H_{0}^{(p)}φ) ) = ∑_{k,l=1}^{d} ((∂_{k}τ, c_{kl}(∂_{l}χ_{n})φ̃) − (τ, (∂_{k}χ_{n})c_{kl}(∂_{l}φ) ))

for all τ ∈ D(H^{(p)∗}) ∩ D(H^{(2)∗}) and φ ∈ D(H_{0}^{(p)} ∩ D(H_{0}^{(p)}), where we indicate by a superscript the L_{p}-space on which the generator acts. Next, for θ ∈ [−π, π]^{d} let W (θ) be the multiplication operator on L_{p}(R^{d}) defined by (W (θ)φ)(x) = e^{iθ·x}φ(x). Set K_{θ} = K_{θ}^{(p)} = W (−θ) HW (θ). Then

(K_{θ}^{(p)∗}τ, χ_{n}φ̃)−(τ, χ_{n}(H_{θ}^{(p)}φ) ) = ∑_{k,l=1}^{d} (((∂_{k}−iθ_{k})τ, c_{kl}(∂_{l}χ_{n})φ̃)−(τ, (∂_{k}χ_{n})c_{kl}((∂_{l}+iθ_{l})φ) ))

for all τ ∈ D(K_{θ}^{(p)∗}) ∩ D(K_{θ}^{(2)∗}) and φ ∈ D(H_{θ}^{(p)}) ∩ D(H_{θ}^{(2)}). Hence

(τ, χ_{n}(λI + H_{θ})^{−1}φ) ) − (τ, (λI + K_{θ})^{−1}(χ_{n}φ̃))
= ∑_{k,l=1}^{d} (((∂_{k} − iθ_{k})(λI + K_{θ}^{∗})^{−1}τ, c_{kl}(∂_{l}χ_{n})((λI + H_{θ})^{−1}φ) )
− ((λI + K_{θ}^{∗})^{−1}τ, (∂_{k}χ_{n})c_{kl}((∂_{l} + iθ_{l})(λI + H_{θ})^{−1}φ) ))

for all τ ∈ L_{q}(R^{d}) ∩ L_{2}(R^{d}), φ ∈ L_{p}(I^{d}) ∩ L_{2}(I^{d}) and λ > 0. Now let φ ∈ L_{p}(I^{d}) ∩ L_{2}(I^{d}), ψ ∈ L_{q;1}(I^{d}) ∩ L_{2;1}(I^{d}) and m ∈ {1, …, d}. Replacing τ by (∂_{m} + iθ_{m})(χ_{n}ψ̃) one computes

((∂_{m} + iθ_{m})ψ, (λI + H_{θ})^{−1}φ) = lim_{n→∞}(2n)^{−d}((∂_{m} + iθ_{m})(χ_{n}ψ̃), (λI + K_{θ})^{−1}(χ_{n}φ̃))

for all λ > 0. On the other hand,

(2n)^{−d} λ^{−1/2}|((∂_{m} + iθ_{m})(χ_{n}ψ̃), (λI + K_{θ})^{−1}(χ_{n}φ̃))| ≤ 2^{d+1} π λ^{−1}∥ψ∥_{q;1}∥φ∥_{p}

uniformly for all n ∈ N and λ > 0. So by the Lebesgue dominated convergence theorem, and the algorithm

A^{−1/2} = π^{−1} ∫_{0}^{∞} dλ λ^{−1/2}(λI + A)^{−1}  (39)

for the inverse square root, one deduces that

((∂_{m} + iθ_{m})ψ, (λI + H_{θ})^{−1/2}φ) = − lim_{n→∞}(2n)^{−d}(χ_{n}ψ̃, (∂_{m} + iθ_{m})(λI + K_{θ})^{−1/2}(χ_{n}φ̃))

for all λ > 0, where we used the boundedness of (∂_{m} +iθ_{m})(λI +K_{θ})^{−1/2} = W (−θ) ∂_{m}(λI + H)^{−1/2}W (θ) on L_{p}(R^{d}), Proposition 3.15. Therefore

|((∂_{m} + iθ_{m})ψ, (λI + H_{θ})^{−1/2}φ)| ≤ ∥(∂_{m} + iθ_{m})(λI + K_{θ})^{−1/2}∥_{p→p}∥ψ∥_{q}∥φ∥_{p}
= ∥∂_{m}(λI + H)^{−1/2}∥_{p→p}∥ψ∥_{q}∥φ∥_{p} .

Hence D((λI + H_{θ})^{1/2}) ⊆ L_{p;1}(I^{d}) and one has the transference inequality

∥R_{k,θ}(λ)∥_{p→p} ≤ ∥R_{k}(λ)∥_{p→p}

for each k ∈ {1, …, d}, λ > 0 and p ∈ ⟨1, ∞⟩ where R_{k,θ}(ε) = (∂_{k} + iθ_{k})(εI + H_{θ})^{−1/2}.

Next,

∥R_{k,θ}(λ)∥_{p→p} ≤ ∥R_{k,θ}(1)∥_{p→p}∥(I + H_{θ})^{1/2}(λI + H_{θ})^{−1/2}∥_{p→p} .

But it follows from [Rob], Lemma II.3.2, and (27) that one has estimates

∥(I + H_{θ})^{1/2}φ∥_{p} ≤ ∥(λI + H_{θ})^{1/2}φ∥_{p} + c^{2}|λ − 1|^{1/2}∥φ∥_{p}

and hence, again by (27),

∥(I + H_{θ})^{1/2}(λI + H_{θ})^{−1/2}∥_{p→p} ≤ 1 + c′(1 + λ)^{1/2}(λ + |θ|^{2})^{−1/2}

uniformly for λ > 0, p ∈ [1, ∞] and θ ∈ [−π, π]^{d}. Combining these estimates one concludes that there is a c > 0 such that

∥R_{k,θ}(λ)∥_{p→p} ≤ c ∥R_{k}(1)∥_{p→p}(λ + 1)^{1/2}(λ + |θ|^{2})^{−1/2}  (40)

uniformly for all k ∈ {1, …, d}, θ ∈ [−π, π]^{d}, λ > 0 and p ∈ ⟨1, ∞⟩.

Step 2 Next we examine the more detailed estimates for small θ starting with θ = 0. The space L_{p}(I^{d}) has a decomposition L_{p}(I^{d}) = C⊕L_{p}^{⊥}(I^{d}), where L_{p}^{⊥}(I^{d}) = {φ ∈ L_{p} : ∫ φ = 0}, and the operators ∂_{m}, H_{0} and H_{0}^{∗} are zero on the component C. Since ∫ φ = (11, φ) and H_{0}^{∗}11 = 0 it follows that L_{p}^{⊥}(I^{d}) is an invariant subspace for H_{0}. Let H_{0}^{⊥} denote the restriction of H_{0} to the space L_{p}^{⊥}. Since Re H_{0}^{⊥} ≥ λ_{1} I > 0 on L_{2}^{⊥}(I^{d}) the restriction S^{⊥} of the semigroup S to L_{2}^{⊥}(I^{d}) has bounds ∥S_{t}^{⊥}∥_{L_{2}^{⊥}→L_{2}^{⊥}} ≤ e^{−λ_{1}t} for all t > 0. Hence by the Gaussian bounds (16) one deduces that ∥S_{t}^{⊥}∥_{L_{1}^{⊥}→L_{∞}^{⊥}} ≤ c e^{−2^{−1}λ_{1}t} for all t ≥ 1. But L_{∞}^{⊥} is continuously embedded in L_{1}^{⊥}, so one has bounds ∥S_{t}^{⊥}∥_{L_{∞}^{⊥}→L_{∞}^{⊥}} ≤ c e^{−2^{−1}λ_{1}t} for all t ≥ 1 and then, by increasing c if necessary, for all t > 0. Hence by duality and interpolation one has bounds ∥S_{t}^{⊥}∥_{L_{p}^{⊥}→L_{p}^{⊥}} ≤ c′ e^{−2^{−1}λ_{1}t} uniform for all t > 0 and p ∈ [1, ∞]. On the other hand,

∥R_{k,0}(λ)∥_{p→p} = ∥∂_{k}(λI + H_{0})^{−1/2}∥_{p→p} = ∥∂_{k}(λI + H_{0}^{⊥})^{−1/2}∥_{L_{p}^{⊥}→L_{p}^{⊥}}

for all p ∈ ⟨1, ∞⟩ and λ > 0. Hence it follows as in the derivation of (40) that there are c, c′ > 0 such that

∥R_{k,0}(λ)∥_{p→p} ≤ c ∥R_{k,0}(1)∥_{p→p}(λ + 1)^{1/2}(λ + λ_{1})^{−1/2} ≤ c′ ∥R_{k}(1)∥_{p→p}(1 + λ_{1}^{−1/2}) (41)

uniformly for all λ > 0. Note that by composition it follows that

sup_{λ>0} ∥∂_{k}(λI + H_{0})^{−1}∂_{l}∥_{p→p} < ∞  (42)

for all k, l ∈ {1, …, d} and p ∈ ⟨1, ∞⟩.

Next we improve the bounds on the transforms with small θ ≠ 0 by use of the θ = 0 bounds. We need the following a priori estimates for p ∈ ⟨1, ∞⟩:

∥∂_{k}(λI + H_{0})^{−1}∂_{l}∥_{p→p} ≤ c_{p} , ∥(λI + H_{θ})^{−1}∥_{p→p} ≤ c (λ + |θ|^{2})^{−1} ,
∥∂_{k}(λI + H_{θ})^{−1}∥_{p→p} ≤ c_{p} (λ + λ_{1})^{−1/2}

uniform for λ > 0 and θ ∈ [−π, π]^{d}. The first of these follows from (42) and the second from Corollary 3.7. The third estimate with θ = 0 follows from (41) because

∥∂_{k}(λI + H_{0})^{−1}∥_{p→p} ≤ ∥R_{k,0}(λ)∥_{p→p}∥(λI + H_{0})^{−1/2}∥_{L_{p}^{⊥}→L_{p}^{⊥}} ≤ c_{p} (λ + λ_{1})^{−1/2} .

The estimate with θ ≠ 0 is more delicate. One has

∥∂_{k}(λI + H_{θ})^{−1}∥_{p→p} ≤ ∥R_{k,θ}(λ)∥_{p→p}∥(λI + H_{θ})^{−1/2}∥_{p→p}
≤ c ∥R_{k}(1)∥_{p→p}(λ + 1)^{1/2}(λ + |θ|^{2})^{−1}

where we have used (40) and the bounds ∥(λI +H_{θ})^{−1/2}∥_{p→p} ≤ c (λ+|θ|^{2})^{−1/2} which follow from (27). Hence for each δ > 0 there is a c_{0} > 0 such that

∥∂_{k}(λI + H_{θ})^{−1}∥_{p→p} ≤ c_{0} ∥R_{k}(1)∥_{p→p}(λ + λ_{1})^{−1/2}

uniformly for λ > 0, θ ∈ [−π, π]^{d} with |θ| > δ, and p ∈ ⟨1, ∞⟩. Now to obtain similar bounds for small θ one starts from the identity

∂_{m}(λI + H_{θ})^{−1} = ∂_{m}(λI + H_{0})^{−1} + ∂_{m}(λI + H_{0})^{−1}(H_{0} − H_{θ})(λI + H_{θ})^{−1}

with

H_{0} − H_{θ} = ∑_{k,l=1}^{d}(iθ_{k}c_{kl}∂_{l} + i∂_{k}c_{kl}θ_{l} − θ_{k}c_{kl}θ_{l}) .  (43)

Then using the first two a priori estimates, and the case θ = 0, one obtains the inequality

sup_{1≤m≤d} ∥∂_{m}(λI + H_{θ})^{−1}∥_{p→p} ≤ c (λ + λ_{1})^{−1/2}  (44)

+ c_{1} (λ + λ_{1})^{−1/2}|θ| sup_{1≤l≤d} ∥∂_{l}(λI + H_{θ})^{−1}∥_{p→p}

+ c_{2} |θ|(λ + |θ|^{2})^{−1} + c_{3} |θ|^{2}(λ + λ_{1})^{−1/2}(λ + |θ|^{2})^{−1}

with c, c_{1}, c_{2}, c_{3} > 0 depending on p, but independent of λ and θ. It follows immediately that for all p ∈ ⟨1, ∞⟩ one may choose c_{p}, δ_{p} > 0 such that

∥∂_{m}(λI + H_{θ})^{−1}∥_{p→p} ≤ c_{p} (λ + |θ|^{2})^{−1/2}

for all λ > 0, θ ∈ [−π, π]^{d} with |θ| ≤ δ_{p}. Substituting these bounds in the right hand side of (44) establishes the third a priori estimate.

Now by (39) one has

|(∂_{k}ψ, ((εI + H_{0})^{−1/2} − (εI + H_{θ})^{−1/2})φ)|

≤ π^{−1} ∫_{0}^{∞} dλ λ^{−1/2}|(∂_{k}ψ, ((λ + ε)I + H_{0})^{−1}(H_{θ} − H_{0})((λ + ε)I + H_{θ})^{−1}φ)| (45)

and after using (43) there are three types of term to estimate in the right hand side of (45). Each can be bounded by use of the above a priori estimates, e.g., the first term is bounded by use of the first two estimates. In each case one finds bounds c ∥ψ∥_{q}∥φ∥_{p} uniform for ε > 0 and θ ∈ [−π, π]^{d}.

One immediately concludes from the foregoing estimates that there is a c > 0 such that

∥R_{k,θ}(ε)∥_{p→p} ≤ ∥R_{k,0}(ε)∥_{p→p} + c

for all ε > 0 and θ ∈ [−π, π]^{d}. Combination with (41) then implies that

sup_{θ∈[−π,π]^{d}} sup_{ε>0} ∥R_{k,θ}(ε)∥_{p→p} < ∞  (46)

for each k ∈ {1, …, d} and p ∈ ⟨1, ∞⟩. Again, by composition, it follows that

sup_{θ∈[−π,π]^{d}} sup_{λ>0} ∥∂_{k}(λI + H_{θ})^{−1}∂_{l}∥_{p→p} < ∞  (47)

for all k, l ∈ {1, …, d} and p ∈ ⟨1, ∞⟩.

Remark 3.16 It follows from (10) that ∥R_{k}(ε)∥_{2→2} is bounded uniformly for ε > 0 and hence

∥∂_{k}φ∥_{2} ≤ c ∥(εI + H)^{1/2}φ∥_{2}

for all φ ∈ D(H^{1/2}). Taking the limit ε → 0 shows that the Riesz transforms are bounded on L_{2}(R^{d}).

At this stage we could deduce the lower bounds of the last statement of Theorem 1.1 for p ∈ ⟨1, 2] and the upper bounds for p ∈ [2, ∞⟩ from known results. Since the kernel of H satisfies Conditions I and II of Theorem 1.1 it suffices, by [AuT], Proposition IV.10, to prove that H and H^{∗} satisfy the Kato property. But this is equivalent to the boundedness of the Riesz transforms of H and H^{∗} on L_{2}(R^{d}) which we have just established.

The proof of boundedness of the Riesz transforms for p > 2 does not seem to follow in the same way from standard results and the example of Kenig, cited above, shows that there is a significant problem. We circumvent this difficulty by an argument that is based on Lemma 2.2 and Proposition 2.3 which works for all p ∈ ⟨1, ∞⟩. This argument relies on estimates on the derivatives of θ ↦ R_{k,θ}(ε).

Step 3 Fix p ∈ ⟨1, ∞⟩. First we consider estimates on the derivatives for θ bounded away from zero. One has

\frac{∂}{∂θ_{l}} R_{k,θ}(ε) = iδ_{kl}(εI + H_{θ})^{−1/2} + (∂_{k} + iθ_{k})\frac{∂}{∂θ_{l}} (εI + H_{θ})^{−1/2} .

But

\frac{∂}{∂θ_{l}} (εI + H_{θ})^{−1/2} = π^{−1} ∫_{0}^{∞} dλ λ^{−1/2} \frac{∂}{∂θ_{l}} ((λ + ε)I + H_{θ})^{−1}
= −π^{−1} ∫_{0}^{∞} dλ λ^{−1/2}((λ + ε)I + H_{θ})^{−1} \frac{∂H_{θ}}{∂θ_{l}} ((λ + ε)I + H_{θ})^{−1}

where

\frac{∂H_{θ}}{∂θ_{l}} = − ∑_{k=1}^{d} (i ∂_{k}c_{kl} + i c_{lk}∂_{k} − (c_{kl} + c_{lk})θ_{k}) .  (48)

Next, by arguments similar to those used in the first part of Step 3, but now using (46) and (47), one deduces that ∥∂R_{k,θ}(ε)/∂θ_{l}∥_{p→p} is uniformly bounded for all ε > 0 and θ ∈ [−π, π]^{d} with |θ| ≥ 4^{−1}θ_{0}, where θ_{0} is as in Subsection 2.2. Then an iteration of this argument establishes that the norms of all the derivatives, ∥∂^{α}R_{k,θ}(ε)/∂θ^{α}∥_{p→p}, have similar uniform bounds.

Next consider the alternative representation R̃_{k,θ}(ε) = V (θ)R_{k,θ}(ε)V (θ)^{−1} of the Riesz transforms. Then R̃_{k,θ}(ε) = ∂_{k}^{θ}(εI + H̃_{θ})^{−1/2} where ∂_{k}^{θ} denotes the partial derivatives with θ-periodic boundary conditions and H̃_{θ} = − ∑_{k,l=1}^{d} ∂_{k}^{θ}c_{kl}∂_{l}^{θ} (see Subsection 2.1). Note that the function θ ↦ ∂_{k}^{θ} is periodic and hence θ ↦ R̃_{k,θ}(ε) is automatically periodic. But since ∂V (θ)/∂θ_{k} = iu_{k} it follows from the previous estimates that the L_{p}-norms of the derivatives ∂^{α}R̃_{k,θ}(ε)/∂θ^{α} are also bounded uniformly for ε > 0 and θ ∈ [−π, π]^{d} with |θ| ≥ 4^{−1}θ_{0}. Hence if ψ ∈ C^{∞}([−π, π]^{d}) is such that ψ(θ) = 1 if |θ| < 4^{−1}θ_{0} and ψ(θ) = 0 if |θ| ≥ 2^{−1}θ_{0} then it follows from Lemma 2.2.II that the operator

R_{ε}(1 − ψ) = Φ^{−1}((2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ (1 − ψ(θ)) R_{k,θ}(ε))Φ
= Φ̃^{−1}((2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ (1 − ψ(θ)) R̃_{k,θ}(ε))Φ̃

is bounded on L_{p}(R^{d}), uniformly for ε > 0.

Step 4 Next we wish to bound derivatives of P_{0}(θ) and to this end we consider H_{θ} for complex θ.

Let

H_{θ}^{(1)} = H_{0} + ∑_{k,l=1}^{d} (i ∂_{k}c_{kl} θ_{l} − θ_{k} c_{kl} θ_{l})

and K^{(θ,1)} the kernel of the semigroup generated by H_{θ}^{(1)}. Then it follows from [ElR3] Theorem 1.1 that there are b, c, κ > 0 such that

max(|K_{t}^{θ}(u; v)|, |K_{t}^{(θ,1)}(u; v)|) ≤ c (1 ∧ t)^{−d/2}e^{−b∥u−v∥^{2}t^{−1}}e^{(κ−1)t}  (49)

uniformly for all u, v ∈ I^{d}, t > 0 and θ ∈ B_{θ_{0}}^{C}. By the Gaussian bounds (49) one has resolvent bounds

∥((λ + κ)I + H_{θ}^{(1)})^{−1}∥_{p→p} ≤ c′(λ + 1)^{−1} and ∥((λ + κ)I + H_{θ})^{−1}∥_{p→p} ≤ c′(λ + 1)^{−1}

uniformly for all λ > 0 and θ ∈ B_{θ_{0}}^{C} and one can deduce from a weak variation of the arguments in Step 2 that D(H_{θ}^{(1)}) ⊂ L_{p;1} for all θ ∈ B_{θ_{0}}^{C}. Next, if V = ∑_{k,l=1}^{d} iθ_{k} c_{kl} ∂_{l} with domain D(V) = L_{p;1} then D(H_{θ}) = D(H_{θ}^{(1)} + V) = D(H_{θ}^{(1)}) ⊂ L_{p;1}.

Then one can view H_{θ} as a perturbation of H_{0} as in Step 2 and the estimates (44) are valid uniformly for complex θ ∈ B_{θ_{0}}^{C} and sufficiently large λ. It follows that there are c, ν > 0 such that

∥∂_{k}((λ + κ)I + H_{θ})^{−1}∥_{p→p} ≤ c (λ + 1)^{−1/2}

uniformly for all λ ≥ ν and θ ∈ B_{θ_{0}}^{C}. Hence by (39) it follows that D(((κ + ν)I + H_{θ})^{1/2}) ⊂ L_{p;1}(I^{d}) and

sup_{k∈{1,...,d}} sup_{θ∈B_{θ_{0}}^{C}} ∥∂_{k}((κ + ν)I + H_{θ})^{−1/2}∥_{p→p} < ∞ .

Using the Gaussian bounds (49) it follows as in Step 1 that D((κI + H_{θ})^{1/2}) ⊂ L_{p;1}(I^{d}) and

sup_{k∈{1,...,d}} sup_{θ∈B_{θ_{0}}^{C}} ∥∂_{k}(κI + H_{θ})^{−1/2}∥_{p→p} < ∞ .

But then it follows as in Step 3 that the map θ ↦ ∂_{k}(κI + H_{θ})^{−1} is holomorphic from B_{θ_{0}}^{C} into L(L_{p}(I^{d})).

Note that the projection P_{0}(θ) is well-defined for all θ ∈ B_{θ_{0}}^{C}. Also the map θ ↦ P_{0}(θ) is holomorphic from B_{θ_{0}}^{C} into L(L_{p}(I^{d})) by Lemma 3.1. Hence by composition the map

θ ↦ ∂_{k}P_{0}(θ) = (κ + λ_{0}(θ))(∂_{k}(κI + H_{θ})^{−1})P_{0}(θ)

is holomorphic from B_{θ_{0}}^{C} into L(L_{p}(I^{d})). By duality, the map θ ↦ P_{0}(θ) ∂_{k} extends to a holomorphic map from B_{θ_{0}}^{C} into L(L_{p}(I^{d})).

Step 5 Introduce the L_{2}(R^{d})-bounded operators R_{ε}^{(1)}(ψ) and R_{ε}^{(2)}(ψ) such that

ΦR_{ε}^{(1)}(ψ)Φ^{−1} = (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ ψ(θ) R_{k,θ}(ε) P_{0}(θ)

and

ΦR_{ε}^{(2)}(ψ)Φ^{−1} = (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ ψ(θ) R_{k,θ}(ε) (I − P_{0}(θ))

and note that

ΦR_{k}(ε)Φ^{−1} = (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ R_{k,θ}(ε) = R_{ε}(1 − ψ) + R_{ε}^{(1)}(ψ) + R_{ε}^{(2)}(ψ) .

It remains to prove that R_{ε}^{(1)}(ψ) and R_{ε}^{(2)}(ψ) are bounded on L_{p}(R^{d}) uniformly for ε ∈ ⟨0, 1].

First we examine R_{ε}^{(1)}(ψ). The function θ ↦ ψ(θ) R_{k,θ}(ε) P_{0}(θ) is clearly periodic, in the sense of Lemma 2.2 since ψ(θ) = 0 if |θ| ≥ 2^{−1}θ_{0}. Note that

R_{k,θ}(ε)P_{0}(θ) = (ε + λ_{0}(θ))^{−1/2}(∂_{k} + iθ_{k})P_{0}(θ) .  (50)

Consider the second term

(ε + λ_{0}(θ))^{−1/2} iθ_{k} P_{0}(θ) = (iθ_{k}λ_{0}(θ)^{−1/2})(λ_{0}(θ)^{1/2} (ε + λ_{0}(θ))^{−1/2})P_{0}(θ) .

The factor (λ_{0}(θ)^{1/2}(ε+λ_{0}(θ))^{−1/2}) is the θ-component in the decomposition of the L_{p}(R^{d})- bounded operator H^{1/2}(εI + H)^{−1/2} acting on P_{0}(B_{θ_{0}})(L_{2}(R^{d}) ∩ L_{p}(R^{d})). Moreover, since θ ↦ λ_{0}(θ) is real-analytic on B_{θ_{0}} and λ_{0}(θ) ≥ μ_{C}|θ|^{2} the factor iθ_{k}λ_{0}(θ)^{−1/2} corresponds to the component of a Z^{d}-invariant multiplier M_{k} satisfying the assumptions of Proposition 2.3. Thus setting P_{0}(ψ) = Φ^{−1}((2π)^{−d} ∫ ψ(θ) P_{0}(θ))Φ the contribution of the second term in (50) to R_{ε}^{(1)} under Φ is the product M_{k} ∘ H^{1/2}(εI + H)^{−1/2} ∘ P_{0}(ψ) of three L_{p}(R^{d})- bounded operators and hence is an L_{p}(R^{d})-bounded operator. Moreover, the L(L_{p}(R^{d})) norm is bounded uniformly in ε > 0.

Now consider the first term in (50),

(ε + λ_{0}(θ))^{−1/2} ∂_{k}P_{0}(θ) = (λ_{0}(θ) (ε + λ_{0}(θ))^{−1})^{1/2}(λ_{0}(θ)^{−1/2} ∂_{k}P_{0}(θ)) .

The first factor is again the component of an L_{p}(R^{d})-bounded operator, bounded uniformly in ε, and it remains to analyze the operator whose components are given by the second factor. But on L_{2}(R^{d}) one has

∫_{B_{θ_{0}}} dθ ∥λ_{0}(θ)^{−1/2}∂_{k}P_{0}(θ)φ_{θ}∥_{2}^{2} ≤ μ_{C}^{−1} ∫_{B_{θ_{0}}} dθ |λ_{0}(θ)|^{−1} Re h_{θ}(P_{0}(θ)φ_{θ})
≤ μ_{C}^{−1} ∫_{B_{θ_{0}}} dθ ∥P_{0}(θ)φ_{θ}∥_{2}^{2} ≤ c ∥φ∥_{2}^{2}

where we have used Lemma 2.4. Hence the corresponding operator is bounded on L_{2}(R^{d}). Since the map θ ↦ ∂_{k}P_{0}(θ) is holomorphic from B_{θ_{0}}^{C} into L(L_{p}(I^{d})) by Step 4 and ∂_{k}P_{0}(0) = 0 it follows that

∂_{k}P_{0}(θ) = ∑_{k=1}^{d} θ_{k}Q_{k}(θ)

where the Q_{k} are holomorphic from B_{θ_{0}}^{C} into L(L_{p}(I^{d})). Hence

λ_{0}(θ)^{−1/2}∂_{k}P_{0}(θ) = ∑_{k=1}^{d} θ_{k} λ_{0}(θ)^{−1/2} Q_{k}(θ) .

But the decomposition of R_{ε}^{(1)}(ψ) is restricted by the factor ψ to components in the ball B_{θ_{0}} and the factor θ_{k} λ_{0}(θ)^{−1/2} corresponds to the component of an L_{p}(R^{d})-bounded multiplier, by Proposition 2.3. Moreover, Q_{k}(θ) corresponds to the component of an L_{p}(R^{d})-bounded operator, by Lemma 2.2.I. Combining all these observations R_{ε}^{(1)}(ψ) is L_{p}(R^{d})-bounded uniformly for ε ∈ ⟨0, 1].

Finally we consider R_{ε}^{(2)}(ψ), the contributions from the large spectrum. Obviously the function θ ↦ ψ(θ) R_{k,θ}(ε) (I − P_{0}(θ)) is periodic, in the sense of Lemma 2.2. One has

(εI + H_{θ})^{−1/2}(I − P_{0}(θ)) = π^{−1} ∫_{0}^{∞} dλ λ^{−1/2} ((λ + ε)I + H_{θ})^{−1}(I − P_{0}(θ))

for all ε > 0 and θ ∈ B_{θ_{0}} and we next analyze the operator in the integrand.

For all λ > 0 and θ ∈ B_{θ_{0}} set

T_{λ}(θ) = (I − P_{0}(θ)) (λI + H_{θ})^{−1} (I − P_{0}(θ)) = (λI + H_{θ})^{−1} (I − P_{0}(θ)) .

First the semigroup S^{θ} leaves the space (I−P_{0}(θ))L_{p}(I^{d}) invariant. Moreover, Re(φ, H_{θ}φ) ≥ μ ∥φ∥_{2}^{2} for all θ ∈ B_{2^{−1}θ_{0}} and φ ∈ (I − P_{0}(θ))(D(H_{θ})), for a suitable μ > 0, by Proposition 2.11. Then by the arguments of Step 2 there is a c > 0 such that ∥S_{t}^{θ}φ∥_{p} ≤ c e^{−2^{−1}μt}∥φ∥_{p} uniformly for all θ ∈ B_{2^{−1}θ_{0}}, t > 0 and φ ∈ (I − P_{0}(θ))L_{p}. Consequently, ∥(λI + H_{θ})^{−1}φ∥_{p} ≤ c (λ + 2^{−1}μ)^{−1}∥φ∥_{p} for all φ ∈ (I − P_{0}(θ))L_{p} and

∥T_{λ}(θ)∥_{p→p} ≤ c′(λ + 2^{−1}μ)^{−1}

uniformly for all λ > 0 and θ ∈ B_{2^{−1}θ_{0}}. By the previous estimates one also has bounds

∥∂_{k}T_{λ}(θ)∥_{p→p} ≤ c′′(λ+1)^{−1/2} , ∥T_{λ}(θ)∂_{l}∥_{p→p} ≤ c′′(λ+1)^{−1/2} and ∥∂_{k}T_{λ}(θ)∂_{l}∥_{p→p} ≤ c′′

uniformly for all λ > 0, θ ∈ B_{2^{−1}θ_{0}} and k, l ∈ {1, …, d}. Secondly, if l ∈ {1, …, d} then with ∂H_{θ}/∂θ_{l} as in (48)

\frac{∂}{∂θ_{l}} T_{λ}(θ) = −\frac{∂P_{0}}{∂θ_{l}} (θ) T_{λ}(θ) − T_{λ}(θ) \frac{∂H_{θ}}{∂θ_{l}} T_{λ}(θ) − T_{λ}(θ) \frac{∂P_{0}}{∂θ_{l}} (θ)

= −\frac{∂P_{0}}{∂θ_{l}} (θ) T_{λ}(θ) − T_{λ}(θ) \frac{∂P_{0}}{∂θ_{l}} (θ) + ∑_{k=1}^{d} T_{λ}(θ)(i ∂_{k}c_{kl} + i c_{lk}∂_{k} − (c_{kl} + c_{lk})θ_{k})T_{λ}(θ)

is a finite linear combination of finite products of the operators

\frac{∂P_{0}}{∂θ_{l}} (θ) , T_{λ}(θ) , T_{λ}(θ)∂_{m} , ∂_{m}T_{λ}(θ)

and the multiplication operators θ_{m}I, c_{ml}I and c_{lm}I. Moreover, each term contains at least one factor T_{λ}(θ). Similarly, ∂(∂_{j}T_{λ}(θ))/∂θ_{l}, ∂(T_{λ}(θ)∂_{k})/∂θ_{l} and ∂(∂_{j}T_{λ}(θ)∂_{k})/∂θ_{l} are a finite linear combination of finite products of the foregoing operators and the operator ∂_{j}T_{λ}(θ)∂_{k}. Hence by induction multiple θ-derivatives of T_{λ}(θ) are also a finite linear combination of finite products of these operators and each term contains at least one factor T_{λ}(θ). But each factor is a bounded operator on L_{p}(I^{d}) and ∥T_{λ}(θ)∥_{p→p} ≤ c(1 + λ)^{−1} uniformly for all λ > 0 and θ ∈ B(2^{−1}θ_{0}). Hence the L_{p}(I^{d})-norm of the θ-derivatives of the integrand of ΦR_{ε}^{(2)}(ψ)Φ^{−1} is uniformly bounded for ε > 0. Therefore ∥R_{ε}^{(2)}(ψ)∥_{p→p} is bounded, uniformly for all ε > 0 by Lemma 2.2.I.

We have now proved that R_{ε}(1 − ψ), R_{ε}^{(1)}(ψ) and R_{ε}^{(2)}(ψ) are all L_{p}(R^{d})-bounded uniformly for ε > 0. It follows that the transforms ∂_{k}(εI + H)^{−1/2} are uniformly L_{p}(R^{d})- bounded and one readily deduces that

c_{p} max_{1≤k≤d} ∥∂_{k}φ∥_{p} ≤ ∥H^{1/2}φ∥_{p}

by taking the limit ε → 0. The complementary bounds

∥H^{1/2}φ∥_{p} ≤ c′_{p} max_{1≤k≤d} ∥∂_{k}φ∥_{p}

then follow by a standard duality argument.

The proof of Theorem 1.1 is complete.

4 Hölder continuous coefficients

If the coefficients of the divergence form operator are Hölder continuous of order ν ∈ ⟨0, 1⟩ then it follows from [AMT] or [ElR1] that the kernel is once-differentiable in both variables and the derivatives are Hölder continuous of order ν. Moreover, these derivatives satisfy Gaussian bounds but with an extra factor which grows polynomially with the t variable. In this section we establish the hierarchy of regularity estimates for periodic systems described in Theorems 1.2, 1.3 and 1.4 which are uniform in time.

4.1 Derivatives of the kernel

In this subsection we prove Theorem 1.2.

Suppose ν ∈ ⟨0, 1⟩ is such that c_{kl} ∈ C^{ν} for all k, l ∈ {1, …, d}. Since the statements concerning the derivative of the kernel with respect to the second variable follow by duality from similar statements with respect to the first variable, we only need to consider the first variable. We set ∂_{k} = ∂/∂x_{k}, the derivative with respect to the first variable in the k-th direction.

The proofs then follow by a variation of the methods of Section 3.

Lemma 4.1 For all θ ∈ C^{d} with |θ| < θ_{0} the projection P_{0}(θ) is bounded from L_{1}(I^{d}) into L_{∞;1}(I^{d}) and ∂_{k}P_{0}(θ) ∈ L(L_{1}(I^{d})), C^{ν}) for all k ∈ {1, …, d}. Moreover, the map θ ↦ ∂_{k}P_{0}(θ) is holomorphic from B_{θ_{0}}^{C} into L(L_{1}(I^{d}), L_{∞}(I^{d})) and from B_{θ_{0}}^{C} into L(L_{1}(I^{d}), C^{ν}) for all k ∈ {1, …, d}.

Proof It follows from [AMT], Theorem 4.15, that there exist b, c > 0 and N ∈ N such that

|(∂_{k}K_{t})(x; y)| ≤ c t^{−1/2} (1 + t)^{N}G_{b,t}(x − y)

for all k ∈ {1, …, d}, x, y ∈ R^{d} and t > 0. Using the identity (15) one sees that K^{θ} is differentiable, the derivative of S_{t}^{θ} maps L_{1}(I^{d}) into L_{∞}(I^{d}) and θ ↦ ∥∂_{k}S_{t}^{θ}∥_{1→∞} is locally uniformly bounded. Then the proof is a variation of the arguments given in the proof of Lemma 3.1.

Finally, the Hölder estimates follow by a small change of the argument starting from the kernel bounds

|(∂_{k}K_{t})(x; y) − (∂_{k}K_{t})(x − h; y)| ≤ c t^{−1/2}(|h|t^{−1/2})^{ν}(1 + t)^{N} G_{b,t}(x − y)

of Theorem 4.15 in [AMT]. □

Obviously ∂_{k}P_{0}(0) = 0 for all k. For k, l ∈ {1, …, d} let P_{kl} ∈ L(L_{1}(I^{d}), L_{∞}(I^{d})) be the partial derivatives of θ ↦ ∂_{k}P_{0}(θ). So

M = sup_{k∈\{1,...,d\}} sup_{{θ∈C^{d} \atop 0<|θ|≤2^{−1}θ_{0}}} |θ|^{−2} ∥∂_{k}P_{0}(θ) − ∑_{l=1}^{d} θ_{l} P_{kl}∥_{1→∞} < ∞ .

Comparison with Corollary 3.4 identifies the operators P_{kl} in terms of the correctors χ_{l}.

Lemma 4.2 If k, l ∈ {1, …, d} then χ_{l} ∈ L_{∞;1}(I^{d}), ∂_{k}χ_{l} ∈ C^{ν} and P_{kl} = i (∂_{k}χ_{l}) P_{0}(0).

Proof It follows from (22) that

P_{0}(θ)11 = φ_{0}(θ) = 11 + i ∑_{l=1}^{d} θ_{l} χ_{l} + O(|θ|^{2})

in the L_{2}-sense. But the map θ ↦ ∂_{k}P_{0}(θ) from B_{θ_{0}}^{C} into L(L_{2}(I^{d}), L_{2}(I^{d})) is also holomorphic for all k ∈ {1, …, d} and, moreover, χ_{l} ∈ L_{2;1}(I^{d}) for all l ∈ {1, …, d}, by Lemma 2.13. Hence

∂_{k}P_{0}(θ)11 = i ∑_{l=1}^{d} θ_{l} ∂_{k}χ_{l} + O(|θ|^{2})

in the L_{2}-sense. On the other hand the map θ ↦ ∂_{k}P_{0}(θ) from B_{θ_{0}}^{C} into L(L_{1}(I^{d}), L_{∞}(I^{d})) is also holomorphic. Therefore ∂_{k}χ_{l} ∈ L_{∞}(I^{d}) for all l ∈ {1, …, d} and hence χ_{k} ∈ L_{∞;1}(I^{d}). Similarly, ∂_{k}χ_{l} ∈ C^{ν}.

It follows from Corollary 3.4 that (ψ, P_{kl}φ) = (ψ, i ∂_{k}χ_{l}) (11, φ) for all φ, ψ ∈ L_{1}(I^{d}), which implies the last statement. □

We are now able to formulate an analogue of Theorems 3.5, 3.8 and 3.9 and Corollary 3.6 for the asymptotic behaviour of ∂_{k}K_{t} for large t. Then the Gaussian bounds for ∂_{k}K_{t} follow as before.

Theorem 4.3 Assume the coefficients are Hölder continuous and periodic with a common period. Then there exists a c > 0 such that

|(∂_{k}K_{t})(x; y) − (∂_{k}K̂_{t})(x − y) − ∑_{l=1}^{d}(∂_{k}χ_{l})(x) (∂_{l}K̂_{t})(x − y)| ≤ c t^{−1} t^{−d/2}

uniformly for all k ∈ {1, …, d}, t ≥ 1 and x, y ∈ R^{d}.

Moreover, for all ε ∈ ⟨0, 1⟩ there exist b, c > 0 such that

|(∂_{k}K_{t})(x; y) − (∂_{k}K̂_{t})(x − y) − ∑_{l=1}^{d}(∂_{k}χ_{l})(x) (∂_{l}K̂_{t})(x − y)| ≤ c t^{−(1−ε)/2} t^{−1/2} G_{b,t}(x − y)

uniformly for all k ∈ {1, …, d}, t ≥ 1 and x, y ∈ R^{d}.

Proof Fix k ∈ {1, …, d}. Since

(x, y) ↦ (∂_{k}K_{t})(x; y) − (∂_{k}K̂_{t})(x − y) − ∑_{l=1}^{d}(∂_{k}χ_{l})(x) (∂_{l}K̂_{t})(x − y)

is the kernel of the operator ∂_{k}S_{t} − ∂_{k}Ŝ_{t} − ∑_{l=1}^{d}(∂_{k}χ_{l}) ∂_{l}Ŝ_{t} it suffices, by Lemma 3.2, to prove that there is a c > 0 such that

∥∂_{k}S_{t}P_{0}(B_{δ}) − ∂_{k}Ŝ_{t}P̂_{0}(B_{δ}) − ∑_{l=1}^{d}(∂_{k}χ_{l}) ∂_{l}Ŝ_{t}P̂_{0}(B_{δ})∥_{1→∞} ≤ c t^{−1} t^{−d/2}  (51)

for all t ≥ 1, where δ = 2^{−1}(θ_{0} ∧ θ̂_{0}). If one differentiates (30) then

Φ∂_{k}S_{t}Φ^{−1} = (2π)^{−d} ∫_{[−π,π]^{d}}^{⊕} dθ (iθ_{k} S_{t}^{θ} + ∂_{k}S_{t}^{θ})

and arguing as in (37) gives

Φ(∂_{k}χ_{l}) ∂_{l}Ŝ_{t}P̂_{0}(B_{δ})Φ^{−1} = (2π)^{−d} ∫_{B_{δ}}^{⊕} dθ e^{−λ̂_{0}(θ)t}iθ_{l} (∂_{k}χ_{l}) P_{0}(0) = (2π)^{−d} ∫_{B_{δ}}^{⊕} dθ e^{−λ̂_{0}(θ)t}θ_{l} P_{kl}

for all t > 0, where we used Lemma 4.2 in the last step. Hence

Φ(∂_{k}S_{t}P_{0}(B_{δ}) − ∂_{k}Ŝ_{t}P̂_{0}(B_{δ}) − ∑_{l=1}^{d}(∂_{k}χ_{l}) ∂_{l}Ŝ_{t}P̂_{0}(B_{δ}))Φ^{−1}

= (2π)^{−d} ∫_{B_{δ}} dθ (iθ_{k}(e^{−λ_{0}(θ)t}P_{0}(θ) − e^{−λ̂_{0}(θ)t}P_{0}(0)) + e^{−λ_{0}(θ)t}(∂_{k}P_{0}(θ) − ∑_{l=1}^{d} θ_{l} P_{kl})

+ (e^{−λ_{0}(θ)t} − e^{−λ̂_{0}(θ)t}) ∑_{l=1}^{d} θ_{l} P_{kl}) .

Arguing as before (51) is valid and the proof of Theorem 4.3 is complete. □

The estimates of Theorem 1.2 on the derivatives (∂_{x_{k}}K_{t})(x; y) now follow from Theorem 4.3 and similar estimates on the derivatives of the Gaussian kernel K̂_{t}.

4.2 Higher order derivatives of the kernel

The hierarchy of smoothness of the kernel together with optimal Gaussian bounds does not usually extend beyond one derivative. The situation is comparable with the scale of genuine Gaussian bounds for derivatives of the kernel of the Laplacian on groups with polynomial growth (see [ERS]).

In this subsection we prove the equivalence of the conditions in Theorem 1.3 by proving the implications III⇒II⇒IV⇒I⇒III and then I⇒V⇒IV. Then we prove Theorem 1.4.

The implication Theorem 1.3.III⇒II is evident. Proof of Theorem 1.3.II⇒IV By (15) in Lemma 4.2 of [ElR2] and Theorem 1.2 it follows that there are b, c > 0 such that

|(∂_{k}K_{t})(x − h; y) − (∂_{k}K_{t})(x; y)| ≤ c t^{−1/2} G_{b,t}(x − y)  (52)

uniformly for all x, y, h ∈ R^{d}, t > 0 and k ∈ {1, …, d} with |h| ≤ t^{1/2}. Interpolation with the bounds of Condition II establishes that for all ε ∈ ⟨0, ν⟩ there are b′, c′ > 0 such that

|(∂_{k}K_{t})(x − h; y) − (∂_{k}K_{t})(x; y)| ≤ c′ t^{−1/2} (|h|t^{−1/2})^{ν−ε} G_{b′,t}(x − y)

uniformly for all x, y, h ∈ R^{d}, t ≥ 1 and k ∈ {1, …, d} with |h| ≤ 1. Then the bounds of Condition IV follow by integration of the kernel.

We next show that the implication IV⇒I is a consequence of Theorem 4.3.

Proof of Theorem 1.3.IV⇒I Suppose Condition IV is valid and that ν ∈ ⟨0, 1⟩, c_{p} > 0 and p ∈ [1, ∞] are such that (5) is satisfied. Fix k ∈ {1, …, d}. Then

∥(I − L(h))∂_{k}S_{t}∥_{p→p} ≤ c_{p} t^{−(1+ν)/2}  (53)

for all t ≥ 1 and h ∈ R^{d} with |h| ≤ 1. We first show that these bounds are also valid with p replaced by 2. The Gaussian bounds of Theorem 1.2 imply that there is a c > 0 such that

∥∂_{k}S_{t}∥_{q→q} ≤ c t^{−1/2}  (54)

uniformly for all q ∈ [1, ∞] and t > 0. Interpolating between (53) and (54) by taking q = 1 if p > 2 and q = ∞ if p < 2 establishes that there are (probably different) c > 0 and ν ∈ ⟨0, 1⟩ such that

∥(I − U(h))∂_{k}S_{t}∥_{2→2} ≤ c t^{−(1+ν)/2}

uniformly for all t ≥ 1 and h ∈ R^{d} with |h| ≤ 1.

It follows from Theorem 4.3 that there is a c′ > 0 such that

∥∂_{k}S_{t} − ∂_{k}Ŝ_{t} − ∑_{l=1}^{d}(∂_{k}χ_{l}) ∂_{l}Ŝ_{t}∥_{2→2} ≤ c′ t^{−(1+ν)/2}

uniformly for all t ≥ 1. So

∥(I − U(h))∂_{k}S_{t} − (I − U(h))∂_{k}Ŝ_{t} − (I − U(h)) ∑_{l=1}^{d}(∂_{k}χ_{l}) ∂_{l}Ŝ_{t}∥_{2→2} ≤ 2c′ t^{−(1+ν)/2}

uniformly for all t ≥ 1. Since Ŝ_{t} satisfies bounds

∥(I − U(h))∂_{l}Ŝ_{t}∥_{2→2} ≤ c′′ t^{−1/2} (|h|t^{−1/2})^{ν}  (55)

uniformly for all t > 0 and h ∈ R^{d} it follows that

∥(I − U(h)) ∑_{l=1}^{d}(∂_{k}χ_{l}) ∂_{l}Ŝ_{t}∥_{2→2} ≤ (c + 2c′ + c′′ |h|^{ν}) t^{−(1+ν)/2} ≤ (c + 2c′ + c′′) t^{−(1+ν)/2}

for all t ≥ 1 and h ∈ R^{d} with |h| ≤ 1. Using the bounds (55) once again, the boundedness of the ∂_{k}χ_{l} and the identity ((U(h) − I)(φψ) = (U(h)φ)((U(h) − I)ψ) + ((U(h) − I)φ)ψ one establishes that there is a c′′′ > 0 such that

∥ ∑_{l=1}^{d} ((I − U(h))(∂_{k}χ_{l}))∂_{l}Ŝ_{t}∥_{2→2} ≤ c′′′ t^{−(1+ν)/2}  (56)

uniformly for all t ≥ 1 and h ∈ R^{d} with |h| ≤ 1.

Fix h ∈ R^{d} with |h| ≤ 1 and set

ψ_{l} = (I − U(h))(∂_{k}χ_{l})

for all l ∈ {1, …, d}. We show that the periodic function ψ_{l} vanishes for all l. (The following elegant argument is due to Nick Dungey.) It follows from (56) and the semigroup property that

∥ ∑_{l=1}^{d} ψ_{l} ∂_{l}K̂_{2t}∥_{2} = ∥ ∑_{l=1}^{d} ψ_{l} ∂_{l}Ŝ_{t}K̂_{t}∥_{2} ≤ c′′′ t^{−(1+ν)/2} ∥K̂_{t}∥_{2} ≤ c t^{−(1+ν)/2} t^{−d/4}

uniformly for all t ≥ 1 and for a suitable c > 0. Then by scaling

∥ ∑_{l=1}^{d} ψ_{l}^{(t)} ∂_{l}K̂_{2}∥_{2} ≤ c t^{−ν/2}

for all t ≥ 1, where ψ_{l}^{(t)}(x) = ψ_{l}(t^{1/2}x). In particular, lim_{t→∞} ∥ ∑_{l=1}^{d} ψ_{l}^{(t)} ∂_{l}K̂_{2}∥_{2} = 0. But

lim_{t→∞} ∥ ∑_{l=1}^{d} ψ_{l}^{(t)} ∂_{l}K̂_{2}∥_{2}^{2} = lim_{t→∞} ∑_{l,m=1}^{d} ∫_{R^{d}} dx \overline{ψ_{m}(t^{1/2}x)} ψ_{l}(t^{1/2}x) \overline{(∂_{m}K̂_{2})(x)} (∂_{l}K̂_{2})(x)

= ∑_{l,m=1}^{d} ∫_{I^{d}} du \overline{ψ_{m}(u)} ψ_{l}(u) · ∫_{R^{d}} dx \overline{(∂_{m}K̂_{2})(x)} (∂_{l}K̂_{2})(x)

= ∫_{I^{d}×R^{d}} du dx | ∑_{l=1}^{d} ψ_{l}(u) (∂_{l}K̂_{2})(x)|^{2}

because \overline{ψ_{m}^{(t)}} ψ_{l}^{(t)} converges in the weak^{∗} topology on L_{∞}(R^{d}) to its average value over I^{d} (see [ZKO] page 5). So ∑_{l=1}^{d} ψ_{l}(u) (∂_{l}K̂_{2})(x) = 0 for almost every (u, x) ∈ I^{d} × R^{d}. Since ψ_{l} and K̂_{2} are continuous by Lemma 4.2 this implies that ∑_{l=1}^{d} ψ_{l}(u) (∂_{l}K̂_{2})(x) = 0 for all (u, x) ∈ I^{d} × R^{d}. Arguing as at the end of the proof of Proposition 3.13 gives (I − U(h))(∂_{k}χ_{l}) = ψ_{l} = 0 for all l ∈ {1, …, d}. This is valid for all k ∈ {1, …, d} and h ∈ R^{d} with |h| ≤ 1. But then ∂_{k}χ_{l} is constant. On the other hand, ∂_{k}χ_{l} ∈ L_{2}^{⊥}(I^{d}), so ∂_{k}χ_{l} = 0. But then χ_{l} is constant and since χ_{l} ∈ L_{2}^{⊥}(I^{d}) one has χ_{l} = 0 for all l ∈ {1, …, d}. Then Condition I follows from Lemma 3.12.I.

Before we can prove the remaining implications of Theorem 1.3 we need some preparation.

Lemma 4.4 If divC = 0 then L_{2;2}(R^{d}) ⊆ D(H) and

Hφ = − ∑_{k,l=1}^{d} c_{kl}∂_{k}∂_{l}φ

for all φ ∈ L_{2;2}(R^{d}).

Proof If ψ ∈ L_{2;1}(R^{d}) and φ ∈ L_{2;2}(R^{d}) then

h(ψ, φ) = ∑_{k,l=1}^{d}(∂_{k}ψ, c_{kl}∂_{l}φ) = − ∑_{k,l=1}^{d}(ψ, c_{kl}∂_{k}∂_{l}φ) + ∑_{l=1}^{d} r(ψ, ∂_{l}φ)

with

r(ψ, χ) = ∑_{k=1}^{d} ((∂_{k}ψ, c_{kl}χ) + (ψ, c_{kl}∂_{k}χ))

for all ψ, χ ∈ L_{2;1}(R^{d}). But then

r(ψ, χ) = ∑_{k=1}^{d} ∫_{R^{d}} dx (∂_{k}(\overline{ψ}χ))(x) c_{kl}(x)

and since divC = 0 one has r(ψ, χ) = 0. □

Since H can be expressed in non-divergent form, and since the coefficients are Hölder continuous, it follows that the kernel K is at least twice-differentiable in the first variable and the derivatives satisfy Gaussian bounds with a possible exponential growth in time (see the remark on page 255 in [Fri]). Explicitly, there are b, c > 0 and ω ≥ 0 such that

|(∂_{k}∂_{l}K_{t})(x; y)| ≤ c t^{−1} e^{ωt}G_{b,t}(x − y)  (57)

and

|(∂_{k}∂_{l}K_{t})(x − h; y) − (∂_{k}∂_{l}K_{t})(x; y)| ≤ c t^{−1}(|h|t^{−1/2})^{ν/2} e^{ωt}G_{b,t}(x − y)  (58)

for all h, x, y ∈ R^{d}, k, l ∈ {1, …, d} and t > 0 with |h| ≤ t^{1/2}. This allows us to extend the conclusions of Lemmas 3.1 and 4.1.

Lemma 4.5 For all θ ∈ B_{θ_{0}}^{C} the projection P_{0}(θ) is bounded from L_{1}(I^{d}) into L_{∞;2}(I^{d}) and ∂_{k}∂_{l}P_{0}(θ) is bounded from L_{1}(I^{d}) into C^{ν/2} for all k, l ∈ {1, …, d}. Moreover, the map θ ↦ ∂_{k}∂_{l}P_{0}(θ) is holomorphic from B_{θ_{0}}^{C} into L(L_{1}(I^{d}), L_{∞}(I^{d})) and from B_{θ_{0}}^{C} into L(L_{1}(I^{d}), C^{ν/2}) for all k, l ∈ {1, …, d}.

Proof Using (15) and (57) one sees that K^{θ} is twice-differentiable and there are b, c, ω > 0 such that

|(∂_{k}∂_{l}K_{t}^{θ})(u; v)| ≤ c (|θ|^{2} + t^{−1})(1 ∧ t)^{−d/2}e^{−b∥u−v∥^{2}t^{−1}}e^{ω(1+(Im θ)^{2})t}

for all θ ∈ C^{d}, t > 0 and u, v ∈ I^{d}. Arguing as in the proof of Lemma 3.1 the L_{∞;2}- statements of the lemma follow. Moreover, the Hölder bounds follow from (58). □

Next we examine the second-order terms in the holomorphic expansion of θ ↦ P_{0}(θ). In general this is complicated but simplifications occur because we are assuming divC = 0. Hence the correctors χ_{m} = 0 by Lemma 3.12. As divC = 0 it also follows that ĉ_{mn} is the average of c_{mn} over I^{d} and one can define

χ_{mn} = 2^{−1}H_{0}^{−1}((c_{mn} + c_{nm}) − (ĉ_{mn} + ĉ_{nm})11)

for all m, n ∈ {1, …, d}, where H_{0}^{−1}: L_{2}^{⊥}(I^{d}) → L_{2}^{⊥}(I^{d}) is as in Subsection 2.3. Then it follows from the proof of Proposition 3.3 that one can identify the second-order term in the expansion of the eigenvector φ_{0}(θ) as φ_{(m,n)} = −χ_{mn} + a_{mn} for some a_{mn} ∈ C. The second-order term φ_{(m,n)}^{†} in the expansion of φ_{0}^{†}(θ) has a more complicated form which we will not require. But one verifies that (φ_{0}^{†}(θ) − (φ_{0}^{†}(θ), 11) 11, φ_{0}(θ)) − 1 = O(|θ|^{3}) and

(ψ, P_{0}^{(mn)}(0)φ) = −(ψ, χ_{mn})(11, φ) + (ψ, 11)(φ_{(m,n)}^{†} − (φ_{θ}^{†}, 11) 11, φ)

for all ψ, φ ∈ L_{2}(I^{d}), where P_{0}^{(mn)}(θ) = ∂^{2}P_{0}(θ)/(∂θ_{m} ∂θ_{n}). As in Corollary 3.4 the constant a_{mn} gives no contribution. Then it follows that

(∂_{k}∂_{l}ψ, P_{0}^{(mn)}(0)φ) = −(∂_{k}∂_{l}ψ, χ_{mn})(11, φ)

for ψ ∈ L_{2;2}(I^{d}) ∩ L_{1;2}(I^{d}) and φ ∈ L_{2} and φ_{(m,n)}^{†} also plays no role. Since θ ↦ ∂_{k}∂_{l}P_{0}(θ) is holomorphic from B_{θ_{0}}^{C} into L(L_{1}(I^{d}), L_{∞}(I^{d})) and from B_{θ_{0}}^{C} into L(L_{1}(I^{d}), C^{ν/2}) it follows that χ_{mn} ∈ L_{∞;2} and ∂_{k}∂_{l}χ_{mn} ∈ C^{ν/2}. Moreover,

∂_{k}∂_{l}P_{0}^{(mn)}(0) = −(∂_{k}∂_{l}χ_{mn})P_{0}(0)

for all k, l, m, n ∈ {1, …, d}.

We extend the χ_{mn} to periodic functions on C(R^{d}).

Proposition 4.6 If divC = 0 then there is a c > 0 such that

|(∂_{k}∂_{l}K_{t})(x; y) − (∂_{k}∂_{l}K̂_{t})(x − y) + ∑_{m,n=1}^{d}(∂_{k}∂_{l}χ_{mn})(x) (∂_{m}∂_{n}K̂_{t})(x − y)| ≤ c t^{−3/2} t^{−d/2}

uniformly for all k, l ∈ {1, …, d}, x, y ∈ R^{d} and t ≥ 1.

Proof Fix k, l ∈ {1, …, d}. Then with δ = 2^{−1}(θ_{0} ∧ θ̂_{0}) one has

Φ(∂_{k}∂_{l}S_{t}P_{0}(B_{δ}) − ∂_{k}∂_{l}Ŝ_{t}P̂_{0}(B_{δ}))Φ^{−1}
= (2π)^{−d} ∫_{B_{δ}} dθ ((iθ_{k} + ∂_{k})(iθ_{l} + ∂_{l})S_{t}^{θ}P_{0}(θ) + θ_{k} θ_{l} Ŝ_{t}^{θ}P_{0}(0))

for all t > 0. If θ ∈ B_{δ} then

(iθ_{k} + ∂_{k})(iθ_{l} + ∂_{l})S_{t}^{θ}P_{0}(θ) + θ_{k} θ_{l} Ŝ_{t}^{θ}P_{0}(0)

= e^{−λ_{0}(θ)t}( − θ_{k} θ_{l} P_{0}(θ) + i θ_{k} ∂_{l}P_{0}(θ) + i θ_{l} ∂_{k}P_{0}(θ) + ∂_{k}∂_{l}P_{0}(θ)) + e^{−λ̂_{0}(θ)t}θ_{k} θ_{l} P_{0}(0)
= ∑_{m,n=1}^{d} e^{−λ̂_{0}(θ)t}θ_{m} θ_{n} ∂_{k}∂_{l}P_{0}^{(mn)}(0) − θ_{k} θ_{l}(e^{−λ_{0}(θ)t}P_{0}(θ) − e^{−λ̂_{0}(θ)t}P_{0}(0))

+ i e^{−λ_{0}(θ)t} θ_{k}∂_{l}P_{0}(θ) + i e^{−λ_{0}(θ)t} θ_{l}∂_{k}P_{0}(θ) + (e^{−λ_{0}(θ)t} − e^{−λ̂_{0}(θ)t})∂_{k}∂_{l}P_{0}(θ)

+ e^{−λ̂_{0}(θ)t}(∂_{k}∂_{l}P_{0}(θ) − ∑_{m,n=1}^{d} θ_{m} θ_{n} ∂_{k}∂_{l}P_{0}^{(mn)}(0))

for all t > 0. But ∂_{k}P_{0}^{(m)}(0) = 0 for all k, m ∈ {1, …, d} by Lemmas 3.12.I and 4.2 since divC = 0. So ∥∂_{k}P_{0}(θ)∥_{1→∞} = O(|θ|^{2}). Then also ∂_{k}∂_{l}P_{0}^{(m)}(0) = 0 for all k, l, m ∈ {1, …, d} and therefore ∥∂_{k}∂_{l}P_{0}(θ)∥_{1→∞} = O(|θ|^{2}). Moreover,

∥∂_{k}∂_{l}P_{0}(θ) − ∑_{m,n=1}^{d} θ_{m} θ_{n} ∂_{k}∂_{l}P_{0}^{(mn)}(0)∥_{1→∞} = O(|θ|^{3}) .

So only the first term gives a big contribution. Since ∂_{k}∂_{l}P_{0}^{(mn)}(0) = −(∂_{k}∂_{l}χ_{mn}) P_{0}(0) this contribution comes from the kernel of the operator −(∂_{k}∂_{l}χ_{mn}) ∂_{m}∂_{n}Ŝ_{t} and the proposition follows by the earlier arguments. □

Now we can complete the proof of the equivalence of the first four conditions of Theorem 1.3.

Proof of Theorem 1.3.I⇒III We first show that one can replace the exponential factor e^{ωt} in the bounds (57) by a factor (1 + t).

It follows from the semigroup property, the estimates (57) and the Gaussian bounds on the kernel that

|(∂_{k}∂_{l}K_{t})(x; y)| ≤ c ∫_{R^{k}} dz G_{b,1}(x − z) G_{b,t−1}(z − y) .

Hence

|(∂_{k}∂_{l}K_{t})(x; y)| ≤ c′ G_{b′,t}(x − y)

uniformly for all t ≥ 1 and x, y ∈ R^{d}.

Since similar bounds are also valid for the kernel K̂ there exist b, c > 0 such that

|(∂_{k}∂_{l}K_{t})(x; y)−(∂_{k}∂_{l}K̂_{t})(x−y)+ ∑_{m,n=1}^{d}(∂_{k}∂_{l}χ_{mn})(x) (∂_{m}∂_{n}K̂_{t})(x−y)| ≤ c (1+t) t^{−1} G_{b,t}(x−y)

uniformly for all t > 0 and x, y ∈ R^{d}. Then by interpolation with the bounds of Proposition 4.6 one deduces that for all ε ∈ ⟨0, 1⟩ there exist b, c > 0 such that

|(∂_{k}∂_{l}K_{t})(x; y) − (∂_{k}∂_{l}K̂_{t})(x − y) + ∑_{m,n=1}^{d}(∂_{k}∂_{l}χ_{mn})(x) (∂_{m}∂_{n}K̂_{t})(x − y)|

≤ c t^{−ε/2} t^{−1} G_{b,t}(x − y) (59)

uniformly for all t ≥ 1 and x, y ∈ R^{d}. This immediately implies Condition III.

Proof of Theorem 1.3.I⇒V Fix p ∈ ⟨1, ∞⟩. It follows from Lemma 4.4 that H can be expressed in non-divergence form. But then D(H) = L_{p;2}(R^{d}) and one has estimates

max_{1≤k,l≤d} ∥∂_{k}∂_{l}φ∥_{p} ≤ c (∥Hφ∥_{p} + ∥φ∥_{p})

for all φ ∈ L_{p;2}(R^{d}). (These estimates follow from similar estimates for operators with constant coefficients, the uniform continuity of the coefficients and a partition of the identity, by standard arguments.) Therefore the operators R_{k,l,ε} = ∂_{k}∂_{l}(εI +H)^{−1} are bounded on L_{p}(R^{d}) for each ε > 0 and again it is necessary to obtain a norm bound uniform for small ε. But

R_{k,l,ε} = ∂_{k}∂_{l} ∫_{0}^{∞} dt e^{−εt}S_{t} = R_{k,l,ε}^{(1)} + R_{k,l,ε}^{(2)}

with

R_{k,l,ε}^{(1)} = ∂_{k}∂_{l} ∫_{0}^{∞} dt e^{−εt}(2e^{−t} − e^{−2t})S_{t} = 2∂_{k}∂_{l}((1 + ε)I + H)^{−1} − ∂_{k}∂_{l}((2 + ε)I + H)^{−1}

and

R_{k,l,ε}^{(2)} = ∫_{0}^{∞} dt e^{−εt}(1 − e^{−t})^{2}∂_{k}∂_{l}S_{t} .

Then ∥R_{k,l,ε}^{(1)}∥_{p→p} is bounded uniformly in ε because H has a bounded H_{∞}-holomorphic functional calculus. Moreover, (59) together with the bounds of III, which are a consequence of I, give bounds

∥∂_{k}∂_{l}S_{t} − ∂_{k}∂_{l}Ŝ_{t} + ∑_{m,n=1}^{d}(∂_{k}∂_{l}χ_{mn}) ∂_{m}∂_{n}Ŝ_{t}∥_{p→p} ≤ c t^{−1} (1 + t)^{−ν/2}

for some ν ∈ ⟨0, 1⟩ and all t > 0. Therefore one estimates easily that ∥R_{k,l,ε}^{(2)}∥_{p→p} is also bounded uniformly in ε.

It follows from these estimates that

max_{1≤k,l≤d} ∥∂_{k}∂_{l}φ∥_{p} ≤ c ∥(εI + H)φ∥_{p}

for all φ ∈ D(H) with c independent of ε. Hence taking the limit ε → 0 one obtains

max_{1≤k,l≤d} ∥∂_{k}∂_{l}φ∥_{p} ≤ c ∥Hφ∥_{p}

for all φ ∈ D(H). But the complementary bounds

c′ ∥Hφ∥_{p} ≤ max_{1≤k,l≤d} ∥∂_{k}∂_{l}φ∥_{p}

are elementary since H is in non-divergence form.

Proof of Theorem 1.3.V⇒IV Suppose Condition V is valid and p ∈ ⟨1, ∞⟩. Then there is a c > 0 such that ∥∂_{k}∂_{l}φ∥_{p} ≤ c ∥Hφ∥_{p} for all k, l ∈ {1, …, d} and φ ∈ L_{p;2}(R^{d}). But the operator H has a bounded H_{∞}-functional calculus on L_{p}(R^{d}). Hence there is a c′ > 0 such that ∥HS_{t}∥_{p→p} ≤ c′ t^{−1} for all t > 0. Then ∥∂_{k}∂_{l}S_{t}φ∥_{p} ≤ c c′t^{−1} ∥φ∥_{p} and Condition IV follows. □

For derivatives higher than second-order on the kernel, or the semigroup, the canonical small time behaviour cannot be expected to extend to large times except in the case that all coefficients are constant and derivatives of all orders are well-behaved.

Proof of Theorem 1.4 The proof of the implication I⇒II is well known.

“II⇒III”. The proof is the similar to the proof of Theorem 1.3.II⇒IV although one more interpolation argument is necessary. It is a consequence of the Duhamel formula that

(∂_{k}φ)(x) = −s^{−1}(φ(x − se_{k}) − φ(x)) − s^{−1} ∫_{0}^{s} du ((∂_{k}φ)(x − ue_{k}) − (∂_{k}φ)(x))

for any continuously differentiable function φ, all x ∈ R^{d}, s > 0 and k ∈ {1, …, d}, where e_{k} is the unit vector in the k-th direction. Setting φ(x) = (∂_{l}K_{t})(x; y) it follows from the bounds (52) and the bounds of Condition II that there are b, c > 0 such that

|(∂_{k}∂_{l}K_{t})(x; y)| ≤ c s^{−1} t^{−1/2} G_{b,t}(x − y) + c s^{−1} ∫_{0}^{s} du t^{−1}(ut^{−1/2})^{ν}t^{−d/2}
= c t^{−(d+1)/2}(s^{−1}e^{−b|x−y|^{2}t^{−1}} + (1 + ν)^{−1}s^{ν}t^{−(1+ν)/2})
≤ 2c t^{−(d+1)/2}e^{−2^{−1}νb|x−y|^{2}t^{−1}}

uniformly for all t ≥ 1 and x, y ∈ R^{d}, if one chooses s = e^{−2^{−1}b|x−y|^{2}t^{−1}} ≤ 1. Hence there are b′, c′ > 0 such that

|(∂_{k}∂_{l}K_{t})(x − h; y) − (∂_{k}∂_{l}K_{t})(x; y)| ≤ c′ (|h|t^{−1/2})^{ν} G_{b′,t}(x − y)

uniformly for all t ≥ 1 and h, x, y ∈ R^{d} with |h| ≤ 1. Interpolation with the bounds of Condition II gives bounds

|(∂_{k}∂_{l}K_{t})(x − h; y) − (∂_{k}∂_{l}K_{t})(x; y)| ≤ c′′ t^{−(1−ε)}(|h|t^{−1/2})^{ν} G_{εb′,t}(x − y)
≤ c′′ t^{−1}(|h|t^{−1/2})^{ν−2ε} G_{εb′,t}(x − y)

uniformly for all ε ∈ ⟨0, ν/2⟩, t ≥ 1 and h, x, y ∈ R^{d} with |h| ≤ 1. Hence III is valid.

“III⇒I”. If Condition III is valid then ∥(I−U(h))(I−U(g))∂_{l}S_{t}∥_{p→p} ≤ c_{p} t^{−(2+ν)/2} |h|^{ν} |g| for all g, h ∈ R^{d}, t ≥ 1 and l ∈ {1, …, d} with |h| ≤ 1. Arguing as in the proof of Theorem 1.3.IV⇒I it follows that (I −U(h))(I −U(g))∂_{k}χ_{l} = 0 for all k, l ∈ {1, …, d} and g, h ∈ R^{d} with |h| ≤ 1. Hence χ_{l} = 0 for all l and divC = 0. Then one can apply Proposition 4.6 and argue as in the proof of the implication in Theorem 1.3.IV⇒I. One deduces that Condition I is valid.

Finally the implication “I⇒IV” is well known and the implication “IV⇒III” follows as in the proof of Theorem 1.3.V⇒IV. □

Remark 4.7 Under certain circumstances the boundedness of the second-order Riesz transforms for periodic operators in non-divergence form can be deduced from the result for divergence form operators by a standard multiplier-factorization procedure (see [BLP] Sections 3.3.3 and 3.4.1, [ZKO] Section 1.3 or [AvL2] Section 1, or [Koz]). Let H_{n} = − ∑_{k,l=1}^{d} c_{kl}∂_{k}∂_{l} be a non-divergence form operator with real coefficients satisfying the ellipticity condition (2). Assume the coefficients c_{kl} are Hölder continuous and periodic with a common period. Then there exist a bounded continuous function m and a strongly elliptic operator in divergence form H̃ with real Hölder continuous periodic coefficients C̃ = (c̃_{kl}), with the same common period, such that ∑_{k=1}^{d} ∂_{k}c̃_{kl} = 0 and H̃ = mH_{n}. Therefore, by Theorem 1.3, there is a c_{p} > 0 such that

c_{p} max_{1≤k,l≤d} ∥∂_{k}∂_{l}φ∥_{p} ≤ ∥H̃φ∥_{p} ≤ ∥m∥_{∞}∥H_{n}φ∥_{p}

for all φ ∈ D(H_{n}) = D(H̃) = L_{p;2}(R^{d}) and p ∈ ⟨1, ∞⟩. Hence the second-order Riesz transforms of H_{n} are bounded. It is evident that one also has bounds

∥H_{n}φ∥_{p} ≤ c′_{p} max_{1≤k,l≤d} ∥∂_{k}∂_{l}φ∥_{p}

for all φ ∈ L_{p;2}(R^{d}).

It is not clear whether the multiplier–factorization procedure can be extended to complex operators or to systems. The standard proof does not immediately generalize.

5 Systems of Operators

Hitherto we considered a single (complex) second-order operator H acting on L_{2}(R^{d}) but in this section we discuss a system of such operators (H_{rs})_{1≤r,s≤n} acting on n copies of L_{2}(R^{d}). Our aim is to explain how one can extend the earlier results to systems satisfying a suitable ellipticity condition.

Assume the coefficients c_{kl}^{rs} ∈ L_{∞}(R^{d}) are continuous and periodic with a common period. Consider the form

h(φ) = ∑_{r,s=1}^{n} ∑_{k,l=1}^{d}(∂_{k}φ_{r}, c_{kl}^{rs} ∂_{l}φ_{s})

with domain D(h) = (L_{2;1}(R^{d}))^{n}. We assume that h satisfies the strong Gårding inequality

Re h(φ) ≥ μ ∑_{r=1}^{n} ∥∇φ_{r}∥_{2}^{2}  (60)

uniformly for all φ ∈ D(h), where μ > 0. This would follow, for example, if the (c_{kl}^{rs}) satisfy the ellipticity condition

ess inf_{x∈R^{d}} Re ∑_{r,s=1}^{n} ∑_{k,l=1}^{d} c_{kl}^{rs}(x) \overline{ξ_{rk}} ξ_{sl} ≥ μ ∑_{r=1}^{n} ∑_{k=1}^{d} |ξ_{rk}|^{2}  (61)

for all ξ_{rk} ∈ C with μ > 0, used by Avellaneda and Lin [AvL1]. Note that (61) is the direct analogue for systems of the earlier condition (2).

The strong Gårding inequality (60) ensures that the operator H = (H_{rs}) associated with the form h is a maximal accretive operator acting on L_{2}(R^{d}, C^{n}). Moreover, since the coefficients c_{kl}^{rs} are automatically uniformly continuous the bounds (60) allow one to extend the analysis of [AuQ] as in [Aus] to obtain bounds analogous to (14).

The decomposition theory described in Section 2 extends naturally to such systems of periodic operators. There are, however, a total of nd variables θ_{rk}, 1 ≤ r ≤ n, 1 ≤ k ≤ d, taking values in [−π, π]. The components in the decomposition of the form h now are

h_{θ}(φ) = ∑_{r,s=1}^{n} ∑_{k,l=1}^{d}((∂_{k} + iθ_{rk})φ_{r}, c_{kl}^{rs} (∂_{l} + iθ_{sl})φ_{s}) .

One can argue that h_{θ} satisfies the analogue of Lemma 2.4, i.e., for each κ ∈ [0, 1] one has estimates

Re h_{θ}(φ) ≥ μ (1 − κ) ∑_{r=1}^{n} ∥∇φ_{r}∥_{2}^{2} + μ ∑_{r=1}^{n} ∑_{k=1}^{d} |θ_{rk}|^{2}∥φ_{r}∥_{2}^{2}  (62)

for all φ ∈ L_{2;1}(I^{d})^{n}, whenever max |θ_{rk}| ≤ κπ. Indeed, if χ_{1}, χ_{2}, … are as in Step 1 of the proof of Theorem 1.1.III in Subsection 3.4 and φ̃_{r}(u − n) = φ_{r}(u) and W_{r}(θ) is the multiplication operator on L_{2}(R^{d}) defined by (W_{r}(θ)φ)(x) = e^{iθ_{r}·x}φ(x) where θ_{r} = (θ_{r1}, …, θ_{rd}) then

Re h_{θ}(φ) = lim_{j→∞}(2j)^{−d} ∑_{r,s=1}^{n} ∑_{k,l=1}^{d}(∂_{k}W_{r}(θ)(χ_{j}φ̃_{r}), c_{kl}^{rs} ∂_{l}W_{s}(θ)(χ_{j}φ̃_{s}))

≥ μ lim_{j→∞}(2j)^{−d} ∑_{r=1}^{n} ∑_{k=1}^{d} ∥∂_{k}W_{r}(θ)(χ_{j}φ̃_{r})∥_{L_{2}(R^{d})}^{2}

= μ lim_{j→∞}(2j)^{−d} ∑_{r=1}^{n} ∑_{k=1}^{d} ∥(∂_{k} + iθ_{rk})(χ_{j}φ̃_{r})∥_{L_{2}(R^{d})}^{2} = μ ∑_{r=1}^{n} ∑_{k=1}^{d} ∥(∂_{k} + iθ_{rk})φ_{r}∥_{L_{2}(I^{d})}^{2}

and then (62) follows as in the proof of Lemma 2.4. The bounds (62) can be used as before to reduce the problem to the analysis of the low lying spectrum of the corresponding operators H_{θ} = (H_{rs;θ}). The main difference in the analysis is a degeneracy of the lowest eigenvalue.

The operators H_{θ} are still a holomorphic family of type (B) and 0 is an eigenvalue of H_{0}. But the multiplicity of this eigenvalue is now n. For example, the vectors (0, …, 11, …, 0) with 11 in the r-th position are all eigenvectors corresponding to the zero eigenvalue. Therefore P_{0}(θ) is still well-defined by (12) for all small |θ| and has rank n. Moreover, θ ↦ P_{0}(θ) is holomorphic near θ = 0. Hence the previous spectral arguments can be applied with very little change. The principal difference occurs in the derivation of analogues of Proposition 3.3 and Corollary 3.4. These results were established by perturbation of the lowest eigenstate of H_{0}. Since this state has an n-fold multiplicity one now has to use degenerate perturbation theory as developed in [Kat] page 100 et seq.. In particular one needs to choose an appropriate basis of the range R(P_{0}(θ)) of the projection P_{0}(θ) to ensure that the matrix elements of the restriction of H_{θ} to the subspace are analytic. We will not enter into further detail. All the results stated in the introduction remain valid for periodic elliptic systems satisfying the strong Gårding inequality with appropriate modifications to take account of the multiple dimensions.

Acknowledgements

The first author is indebted to the Centre of Mathematics and its Applications at the ANU for hospitality and support whilst this work was carried out. All three authors would like to thank Nick Dungey for simplifying the proof of Theorem 1.3 and Pascal Auscher for helpful comments. Finally the authors are grateful to the referee for various percipient remarks including the observation that the Riesz transforms for periodic operators with continuous coefficients should be bounded. Our original proof required Hölder continuity.

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