IMAGINARY POWERS OF LAPLACE OPERATORS

ADAM SIKORA AND JAMES WRIGHT

Abstract. We show that if L is a second-order uniformly elliptic operator in divergence form on R^{d}, then C_{1}(1 + |α|)^{d/2} ≤ ∥L^{iα}∥_{L^{1}→L^{1,∞}} ≤ C_{2}(1 + |α|)^{d/2}. We also prove that the upper bounds remain true for any operator with the finite speed propagation property.

1. Introduction. Assume that a_{ij} ∈ C^{∞}(R^{d}), a_{ij} = a_{ji} for 1 ≤ i, j ≤ d and that κI ≤ (a_{ij}) ≤ τI for some positive constants κ and τ. We define a positive self-adjoint operator L on L^{2}(R^{d}) by the formula

L = − ∑ ∂_{i}a_{ij}∂_{j}. (1)

We refer readers to [8] for the precise definition and basic properties of L. In particular, L admits a spectral resolution E(t) and we can define the operator L^{iα} by the formula

L^{iα} = ∫_{0}^{∞} t^{iα}dE(t).

By spectral theory ∥L^{iα}∥_{L^{2}→L^{2}} = 1. It is well known that L^{iα} falls within the scope of classical Calderón-Zygmund theory (as described in [3] or [22]) and so it extends to a bounded operator on L^{p}, 1 < p < ∞, and is also weak type (1,1). The main aim of this paper is to obtain the sharp estimate for the weak type (1, 1) norm of L^{iα} in terms of α.

The study of imaginary powers of operators is an important part of the theory of operators of type ω with H^{∞} functional calculus, see e.g., [6], [9] and [17]. What is perhaps more interesting and relevant from the point of view of this paper is that the weak type (1, 1) norm of imaginary powers of self-adjoint operators can play a central role in the theory of spectral multipliers. See [5] and [15]. Imaginary powers of Laplace operators on compact Lie groups were also investigated in [20]. Theorem 2 below applied to Laplace operators on compact Lie groups gives the sharp endpoint result of Theorem 3 in [20], pp. 58. See also Corollary 4 of [20], pp. 121.

However, the starting point for this paper is the following observation from [2]. If we denote the weak type (1,1) norm of an operator T on a measure space (X, μ) by ∥T ∥_{L^{1}→L^{1,∞}} = sup λ μ({x ∈ X : |T f(x)| > λ}) where the supremum is taken over λ > 0 and functions f with L^{1}(X) norm less than one, then for the standard Laplace operator on R^{d},

C_{1}(1 + |α|)^{d/2} ≤ ∥(−Δ_{d})^{iα}∥_{L^{1}→L^{1,∞}} ≤ C_{2}(1 + |α|)^{d/2} log(1 + |α|). (2)

The classical Hörmander multiplier theorem (see [13]) states that a multiplier operator T_{m} on R^{d} with multiplier m satisfies

∥T_{m}∥_{L^{1}→L^{1,∞}} ≤ C_{s} sup_{t>0} ∥η(·)m(t·)∥_{H_{s}} ≤ A (3)

for any s > d/2 and any η ∈ C_{c}^{∞}(R_{+}) not identically zero. Here H_{s} is the Sobolev space of order s on R^{d}. Since the Sobolev norm in (3) behaves like (1 + |α|)^{s} for the multiplier m(x) = |x|^{iα} of (−Δ)^{iα}, (2) shows that the exponent d/2 in Hörmander’s theorem is sharp. Furthermore, if (3) is satisfied with A < ∞, then the distribution K = m̂ agrees with a locally integrable function away from the origin which satisfies

1991 Mathematics Subject Classification. Primary 42B15; Secondary 35P99.

Key words and phrases. Spectral multiplier, imaginary powers.

I(B) = sup_{y≠0} ∫_{|x|≥B|y|} |K(x − y) − K(x)| dx ≤ A (4)

for B ≥ 2 and Hörmander’s theorem actually shows that the weak type (1,1) norm of T_{m} is bounded by I(B) + ∥m∥_{L^{∞}}^{2} + B^{d}. One can easily compute that for the convolution kernel K of (−Δ)^{iα}, the integral I(B) is bounded above and below by (1 + |α|)^{d/2} log(1 + |α|/B). Hence Hörmander’s theorem gives the upper bound in (2). The lower bound is a simple consequence of the explicit formula for the kernel K of (−Δ)^{iα}. See for example, [21] pp. 51-52.

The main observation of this paper is to note that there is a slight improvement of the bound I(B) + ∥m∥_{L^{∞}}^{2} + B^{d} to I(B) + (∥m∥_{L^{∞}}^{2}B^{d})^{1/2}. This can be achieved either by using C. Fefferman’s ideas in [11] of exploiting more information of L^{2} bounds or by varying the level of the Calderón - Zygmund decomposition and optimising. Hence we will be able to remove the log term in (2). We will show that this more precise estimate holds for a general class of operators.

Theorem 1. Suppose that L is defined by (1). Then

C_{1}(1 + |α|)^{d/2} ≤ ∥L^{iα}∥_{L^{1}→L^{1,∞}} ≤ C_{2}(1 + |α|)^{d/2} (5)

for all α ∈ R.

Proof of the lower bound. We begin with some known estimates for the kernel p_{t}(x, y) of the heat operator e^{−tL} associated to L. Firstly, this kernel satisfies Gaussian bounds

C_{1} \frac{1}{t^{d/2}} e^{−b_{1}ρ^{2}(x,y)/t} ≤ p_{t}(x, y) ≤ C_{2} \frac{1}{t^{d/2}} e^{−b_{2}ρ^{2}(x,y)/t} (6)

(see [8]) for some positive constants C_{1}, C_{2}, b_{1} and b_{2} and where ρ(x, y) denotes the geodesic distance between x and y given by the Riemannian metric (a_{i,j}). In this setting of uniform ellipticity, κ|x − y| ≤ ρ(x, y) ≤ τ|x − y|. Secondly, from the construction of a parametrix for the heat equation with respect to L (either via Hadamard’s construction, see §17.4 of [14] , or using pseudodifferential operator techniques, see chapter 7, §13 of [23]), we have for each y ∈ R^{d}, a ball B(y, r) such that for x ∈ B(y, r) and 0 < t < 1,

|p_{t}(x, y) − (det a_{ij}(y))^{−1/2}(4πt)^{−d/2}e^{−ρ^{2}(x,y)/4t}| ≤ Ct^{1/2}t^{−d/2}. (7)

Here we are using the fact that p_{t} is symmetric, p_{t}(x, y) = p_{t}(y, x). ¿From (6) and (7), we have for x ∈ B(y, r) the bound

|p_{t}(x, y) − (det a_{ij}(y))^{−1/2}(4πt)^{−d/2}e^{−ρ^{2}(x,y)/4t}| ≤ Ct^{1/4}t^{−d/2} exp (−b′ρ(x, y)^{2}/t)

which translates into a bound for the kernel K_{L^{iα}} of L^{iα} since the functional calculus for L gives us the relationship

L^{iα} = Γ(−iα)^{−1} ∫_{0}^{∞} t^{−iα−1}e^{−tL}dt

for α ≠ 0. Thus for x ∈ B(y, r),

|K_{L^{iα}}(x, y) − (det a_{ij}(y))^{−1/2}4^{iα}π^{−d/2}γ(α)ρ(x, y)^{−d−i2α}| ≤ C|Γ(−iα)|^{−1}ρ(x, y)^{−d+1/2} (8) where γ(α) = Γ(iα + d/2)/Γ(−iα). Using (8) with y = 0 we obtain for λ large enough

μ(\{|K_{L^{iα}}(x, 0)| ≥ λ\})≥μ(\{C_{1}|γ(α)|ρ^{−d}(x, 0) ≥ 2λ\})−μ(\{C_{2}|Γ(−iα)|ρ^{−d+ \frac{1}{2}} (x, 0) ≥ λ\}) = μ(B(0, (2C_{1}|γ(α)|/λ)^{1/d})) − μ(B(0, (C_{2}|Γ(−iα)|/λ)^{1/(d−1/2)})) ≥ C′|γ(α)|/λ.

Here μ is Lebesgue measure and the sets above have the further restriction that x ∈ B(0, r). Since K_{L^{iα}} is smooth away from the diagonal, we see that L^{iα}ϕ_{δ}(x) tends to K_{L^{iα}}(x, 0) as δ → 0 for any x ≠ 0 and any approximation of the identity {ϕ_{δ}}. Hence the above estimate shows that the weak type (1,1) norm of L^{iα} is bounded below by |γ(α)|=|Γ(iα + d/2)/Γ(−iα)|∼(1 + |α|)^{\frac{d}{2}} (see [10]).

The upper bound in Theorem 1 holds in a much more general setting which we describe now. Assume that (X, μ, ρ) is a space with measure μ and metric ρ. If ∥P ∥_{L^{2}→L^{∞}} < ∞ then we can define the kernel K_{P} of the operator P by the formula

⟨P (ψ), ϕ⟩ = ∫ P (ψ)\overline{ϕ}dμ = ∫ K_{P} (x, y)ψ(x)\overline{ϕ(y)}dμ(x)dμ(y).

Note that sup_{x} ∥K_{P} (x, ·)∥_{L^{2}} = ∥P ∥_{L^{2}→L^{∞}}. Next, we say that

supp K_{P} ⊂ {(x, y) ∈ X^{2} : ρ(x, y) ≤ r} (9)

if ⟨P (ψ), ϕ⟩ = 0 for every ϕ, ψ ∈ L^{2} and every r_{1} + r_{2} + r < ρ(x′, y′) such that ψ(x) = 0 for ρ(x, x′) > r_{1} and ϕ(x) = 0 for ρ(x, y′) > r_{2}. This definition (9) makes sense even if ∥P ∥_{L^{2}→L^{∞}} = ∞. Now if L is a self-adjoint positive definite operator acting on L^{2}(μ) then we say that it satisfies the finite speed propagation property of the corresponding wave equation if

supp K_{C_{t}(\sqrt{L})} ⊂ \{(x, y) ∈ X^{2} : ρ(x, y) ≤ t\}, (10)

where C_{t}(\sqrt{L}) = ∫ cos(t\sqrt{λ}) dE(λ).

Theorem 2. Suppose that L satisfies (10). Next assume that

∥ exp(−tL)∥_{L^{2}→L^{∞}}^{2} ≤ C_{1}V_{d,D}(t^{1/2})^{−1} ≤ Cμ(B(x, t^{1/2}))^{−1} ≤ C_{2}V_{d,D}(t^{1/2})^{−1} (11)

for all t > 0 and x ∈ X, where B(x, t) is a ball with radius t centred at x and

V_{d,D}(t) = \{ \begin{matrix}t^{d} for t ≤ 1 \\ t^{D} for t > 1 \end{matrix}

for d, D ≥ 0. Then

∥L^{iα}∥_{L^{1}→L^{1,∞}} ≤ C_{2}(1 + |α|)^{max(d,D)/2}

for all α ∈ R.

We remark that (10) and (11) are equivalent to having Gaussian upper bounds on the heat kernel and the associated volume growth on balls. See [18]. Furthermore, the upper bound in Theorem 1 follows from Theorem 2. Indeed, if X = R^{d}, ρ(x, y) = τ|x − y| and μ is Lebesgue measure then it is well known (see e.g. [8] and [19]) that (11) and (10) hold. We are going to prove Theorem 2 only in the case d = D. The argument for the other cases is similar.

2. Preliminaries. The following lemma is a very simple but useful consequence of (10).

Lemma 1. Assume that L satisfies (10) and that F̂ is a Fourier transform of an even bounded Borel function F with supp F̂ ⊂ [−r, r]. Then

supp K_{F (\sqrt{L})} ⊂ \{(x, y) ∈ X^{2} : ρ(x, y) ≤ r\}.

Proof. If F is an even function, then by the Fourier inversion formula,

F (\sqrt{L}) = \frac{1}{2π} ∫_{−∞}^{+∞} F̂(t)C_{t}(\sqrt{L}) dt.

But since supp F̂ ⊂ [−r, r],

F (\sqrt{L}) = \frac{1}{2π} ∫_{−r}^{r} F̂(t)C_{t}(\sqrt{L}) dt

and Lemma 1 follows from (10).

Lemma 2. Let ϕ ∈ C_{c}^{∞}(R) be even, ϕ ≥ 0, ∥ϕ∥_{L^{1}} = 1, supp(ϕ) ⊂ [−1, 1], and set ϕ_{r}(x) = 1/r ϕ(x/r) for r > 0. Let Φ denote the Fourier transform of ϕ and Φ^{r} denote the Fourier transform of ϕ_{r}. If (11) and (10) hold, then the kernel K_{Φ^{r}(\sqrt{L})} of the self-adjoint operator Φ^{r}(\sqrt{L}) satisfies

supp K_{Φ^{r}(\sqrt{L})} ⊂ \{(x, y) ∈ X^{2}; ρ(x, y) ≤ r\} (12)

and

|K_{Φ^{r}(\sqrt{L})}(x, y)| ≤ C r^{−d} (13)

for all r > 0 and x, y ∈ X.

Proof. (12) follows from Lemma 1. For any m ∈ N and r > 0, we have the relationship

(I + rL)^{−m} = \frac{1}{m!} ∫_{0}^{∞}e^{−rtL}e^{−t}t^{m−1} dt

and so when m > d/4, (11) implies

∥(I + rL)^{−m}∥_{L^{2}→L^{∞}} ≤ \frac{1}{m!} ∫_{0}^{∞} ∥ exp (−rtL)∥_{L^{2}→L^{∞}}e^{−t}t^{m−1}dt ≤ C_{1} r^{−d/4} (14)

for all r > 0. Now ∥(I + r^{2}L)^{−m}∥_{L^{1}→L^{2}} = ∥(I + r^{2}L)^{−m}∥_{L^{2}→L^{∞}} and so

∥Φ^{r}(\sqrt{L})∥_{L^{1}→L^{∞}} ≤ ∥(I + r^{2}L)^{2m}Φ^{r}(\sqrt{L})∥_{L^{2}→L^{2}} ∥(I + r^{2}L)^{−m}∥_{L^{2}→L^{∞}}^{2}.

The L^{2} operator norm of the first term is equal to the L^{∞} norm of the function (1 + r^{2}|t|)^{2m}Φ(r\sqrt{|t|}) which is uniformly bounded in r > 0 and so (13) follows by (14).

Next we recall the Calderón-Zygmund decomposition in the general setting of spaces of homogeneous type (see e.g. [3] or [22]) .

Lemma 3. There exists C such that, given f ∈ L^{1}(X, μ) and λ > 0, one can decompose f as

f = g + b = g + ∑ b_{i}

so that 1. |g(x)| ≤ Cλ, a.e. x and ∥g∥_{L^{1}} ≤ C∥f∥_{L^{1}}.

2. There exists a sequence of balls B_{i} = B(x_{i}, r_{i}) such that the support of each b_{i} is contained in B_{i} and

∫ |b_{i}(x)|dμ(x) ≤ Cλμ(B_{i}).

3. ∑ μ(B_{i}) ≤ C \frac{1}{λ} ∫ |f(x)| dμ(x).

4. There exists k ∈ N such that each point of X is contained in at most k of the balls B(x_{i}, 2r_{i}).

We are now in a position to prove Theorem 2.

3. Proof of Theorem 2. The proof follows closely the line of argument in [1] (which of course generalises to this setting). We are out to prove

λ μ(\{x ∈ X : |L^{iα}f(x)| ≥ λ \}) ≤ C(1 + |α|)^{\frac{d}{2}} ∥f∥_{L^{1}}.

As usual we start by decomposing f into g + ∑ b_{i} at the level of λ according to Lemma 3. We will follow the idea of C. Fefferman [11] of using more information of the L^{2} operator norm (in our case, ∥L^{iα}∥_{L^{2}→L^{2}} = 1) by smoothing out the bad functions b_{i} at a scale smaller than the size of it’s support and considering this part of the good function where L^{2} estimates can be used (see also [4]). In our case for each b_{i}, consider Φ^{s_{i}}(\sqrt{L})b_{i} where s_{i} = θr_{i}, θ = (1 + |α|)^{− \frac{1}{2}} , and let G = g + ∑ Φ^{s_{i}}(\sqrt{L})b_{i} be the modified good function. Hence f = G + B where B = ∑(I − Φ^{s_{i}}(\sqrt{L}))b_{i} and we write

λ μ({|L^{iα}f(x)| ≥ λ}) ≤ λ μ({|L^{iα}G(x)| ≥ λ/2}) + λ μ({|L^{iα}B(x)| ≥ λ/2}). (15)

The first term is less than 4/λ ∥L^{iα}G∥_{L^{2}}^{2} ≤ 4/λ ∥G∥_{L^{2}}^{2}. However, according to Lemma 2,

|Φ^{s_{i}}(\sqrt{L})b_{i}(x)| ≤ ∫ |K_{Φ^{s_{i}}(\sqrt{L})}(x, y)b_{i}(y)| dμ(y) ≤ C (θr_{i})^{−d}∥b_{i}∥_{L^{1}}11_{B(x_{i},2r_{i})}

and therefore by Lemma 3, |G(x)| ≤ Cθ^{−d}λ for a.e., x. Using Lemma 2 again which shows that the L^{p} → L^{p} operator norms of Φ^{r}(\sqrt{L}) are uniformily bounded in r > 0, we also have that ∥G∥_{L^{1}} ≤ ∥g∥_{L^{1}} + C ∑ ∥Φ^{s_{i}}(\sqrt{L})b_{i}∥_{L^{1}} ≤ ∥g∥_{L^{1}} + C ∑ ∥b_{i}∥_{L^{1}} ≤ C∥f∥_{L^{1}}. Therefore the first term in (15) is bounded by (1 + |α|)^{\frac{d}{2}} ∥f∥_{L^{1}}.

Since μ(∪B(x_{i}, θ^{−1}r_{i})) ≤ Cθ^{−d} ∑ μ(B_{i}) ≤ C(1+|α|)^{\frac{d}{2}} ∥f∥_{L^{1}}/λ, then to bound the second term in (15), it suffices to show

∫_{x∉∪B_{i}^{∗}} |L^{iα}B(x)| dμ(x) ≤ C(1 + |α|)^{\frac{d}{2}} ∥f∥_{L^{1}} (16)

where B_{i}^{∗} = B(x_{i}, θ^{−1}r_{i}). Let H^{α}(t) = |t|^{2iα} so that L^{iα}B(x) = ∑ H^{α}(1 − Φ^{s_{i}})(\sqrt{L})b_{i}(x) and therefore the left side of (16) is less than

∑_{i} ∫_{x∉∪_{j}B_{j}^{∗}} |∫ K_{H^{α}(1−Φ^{s_{i}})(\sqrt{L})}(x, y)b_{i}(y) dμ(y)| dμ(x)

≤ ∑_{i} ∫ |b_{i}(y)| ∫_{x∉B_{i}^{∗}} |K_{H^{α}(1−Φ^{s_{i}})(\sqrt{L})}(x, y)| dμ(x) dμ(y).

Since F (L)^{∗} = \overline{F} (L), we may interchange the roles of x and y, and so (16) will follow from Lemma 3 once we establish

sup_{x,i} ∫_{ρ(x,y)≥ θ^{−1}r_{i}} |K_{H^{α}(1−Φ^{s_{i}})(\sqrt{L})}(x, y)| dμ(y) ≤ C (1 + |α|)^{\frac{d}{2}} . (17)

We now fix x ∈ X and i. Let η ∈ C_{c}^{∞}(R) be an even function supported in {t ∈ R : 1 ≤ |t| ≤ 4} such that

∑_{n=−∞}^{∞} η(2^{−n}t) = 1 for all t ≠ 0.

We put H_{n}^{α}(t) = η(2^{−n}t)H^{α}(t) so that

H^{α}(1 − Φ^{s_{i}})(\sqrt{L}) = ∑_{n} H_{n}^{α}(1 − Φ^{s_{i}})(\sqrt{L}).

Thus

∫_{y∉B_{i}^{∗}} |K_{H^{α}(1−Φ^{s_{i}})(\sqrt{L})}(x, y)| dμ(y) ≤ ∑_{n} ∫_{y∉B_{i}^{∗}} |K_{H_{n}^{α}(1−Φ^{s_{i}})(\sqrt{L}))}(x, y)| dμ(y) (18)

and we will estimate each term in the sum on the right side in terms of n and i, uniformly in x ∈ X.

Let k_{o} = [d/2] + 1 so that

∫_{y∉B_{i}^{∗}} (1 + 2^{n}ρ(x, y))^{−2k_{o}}dμ(y) ≤ C ∫_{θ^{−1} r_{i}}^{∞} (1 + 2^{n}r)^{−2k_{0}}r^{d−1}dr ≤ C2^{−2nk_{o}}(θ^{−1}r_{i})^{d−2k_{o}}

and therefore by the Cauchy-Schwarz inequality,

∫_{y∉B_{i}^{∗}} |K_{H_{n}^{α}(1−Φ^{s_{i}})(\sqrt{L}))}(x, y)| dμ(y) (19)

≤ C 2^{−nk_{o}}(θ^{−1}r_{i})^{\frac{d}{2} −k_{o}}( ∫_{ρ(x,y)≥θ^{−1}r_{i}} |K_{H_{n}^{α}(1−Φ^{s_{i}})(\sqrt{L})}(x, y)|^{2}(1 + 2^{n}ρ(x, y))^{2k_{o}}dμ(y))^{1/2}.

We break up the integral on the right side of (19) where 2^{n}ρ(x, y) is roughly constant and consider

∑_{2^{j}≥2^{n}r_{i}θ^{−1}} 2^{2jk_{o}} ∫_{2^{j−1−n}<ρ(x,y)≤2^{j−n}} |K_{H_{n}^{α}(1−Φ^{s_{i}})(\sqrt{L})}(x, y)|^{2} dμ(y). (20)

Fix a nonnegative even φ ∈ C_{c}^{∞}(R) such that φ = 1 on [−1/4, 1/4] and φ = 0 on R \ [−1/2, 1/2] . Then the Fourier transforms of H_{n}^{α}(1 − Φ^{s_{i}}) and H_{n}^{α}(1 − Φ^{s_{i}}) ∗ (δ − φ̂_{2^{n−j}}) agree on \{ξ : |ξ| ≥ 2^{j−1−n}\} and so by Lemma 1, the kernels of H_{n}^{α}(1 − Φ^{s_{i}})(\sqrt{L}) and H_{n}^{α}(1 − Φ^{s_{i}}) ∗ (δ − φ̂_{2^{n−j}})(\sqrt{L}) agree on the set \{(x, y) ∈ X^{2} : ρ(x, y) ≥ 2^{j−1−n} \}. Here δ denotes the Dirac mass at 0. For each j, the integrals in (20) satisfy the bound

∫_{2^{j−1−n}<ρ(x,y)≤2^{j−n}} |K_{H_{n}^{α}(1−Φ^{s_{i}})(\sqrt{L})}(x, y)|^{2} dμ(y) ≤ ∥K_{F_{n,j}^{α}(\sqrt{L})}∥_{L^{2}→L^{∞}}^{2}

where we are defining F_{n,j}^{α}(t) = H_{n}^{α}(1 − Φ^{s_{i}}) ∗ (δ − φ̂_{2^{n−j}})(t). So by (14), the right side of this inequality is bounded by ∥(I + 2^{−2n}L)^{m}F_{n,j}^{α}(\sqrt{L})∥_{L^{2}→L^{2}}^{2} 2^{nd} as long as m > d/4. Everything then comes down to estimating the L^{∞} norm of (1 + 2^{−2n}t^{2})^{m}F_{n,j}^{α}(t). We make the following claim.

Claim: For each j, n and m > d/4,

(1 + 2^{−2n}t^{2})^{m}|F_{n,j}^{α}(t)| ≤ C_{m}|α|^{k_{o}}2^{−jk_{o}} min(1, (2^{n}r_{i}θ)^{2}) min(1, |α|2^{−j})

uniformly in t ∈ R.

The claim shows that

∥K_{F_{n,j}^{α}}(\sqrt{L})∥_{L^{2}→L^{∞}} ≤ C |α|^{k_{o}}2^{−jk_{o}}2^{\frac{nd}{2}} min(1, (2^{n}r_{i}θ)^{2}) min(1, |α|2^{−j})

and hence the sum in (20) is bounded by

|α|^{2k_{o}}2^{nd}min^{2}(1, (2^{n}r_{i}θ)^{2}) ∑_{2^{j}≥2^{n}r_{i}θ^{−1}} min^{2}(1, |α|2^{−j}) ≤ |α|^{2k_{o}}2^{nd}min^{2}(1, (2^{n}r_{i}θ)^{2}) log(2+\frac{|α|}{2^{n}r_{i}θ^{−1}} ).

Recall that θ and α are related so that θ|α| = |α|/(1 + |α|)^{\frac{1}{2}} ≤ θ^{−1}. Plugging this into (19) gives

∫_{y∉B_{i}^{∗}} |K_{H_{n}^{α}(1−Φ^{s_{i}})(\sqrt{L})}(x, y)dμ(y) ≤ θ^{−d}(2^{n}r_{i}θ)^{\frac{d}{2} −k_{o}} min(1, (2^{n}r_{i}θ)^{2}) log(2 + \frac{1}{2^{n}r_{i}θ})

and this makes the sum in (18) bounded by θ^{−d} = (1 + |α|)^{\frac{d}{2}} , proving (17) and hence Theorem 2.

Proof of the Claim. If G_{n}(t) = H_{n}^{α}(t)(1 − Φ^{s_{i}}(t)), then F_{n,j}^{α}(t) = 2^{(n−j)k_{o}}G_{n}^{(k_{o})} ∗ ψ̂_{2^{n−j}}(t) where ψ(ξ) = ξ^{−k_{o}}(1 − φ(ξ)) (and so ψ̂ is continuous, rapidly decreasing and has vanishing moments, ∫ t^{ℓ}ψ̂(t)dt = 0, ℓ = 0, 1, 2, ... ). Hence

F_{n,j}^{α}(t) = 2^{(n−j)k_{o}} ∫_{R} [G_{n}^{(k_{o})}(t − s) − G_{n}^{(k_{o})}(t)]ψ̂_{2^{n−j}}(s) ds

= 2^{(n−j)k_{o}} ∫_{R} [G_{n}^{(k_{o})}(t − 2^{n−j}s) − G_{n}^{(k_{o})}(t)]ψ̂(s) ds.

However G_{n}(t) = η(2^{−n}t)|t|^{2iα}(1 − Φ(s_{i}t)) and thereby each time we take a derivative, we gain a factor of 2^{−n}. G_{n}^{(k_{o})}(t) is thus a finite sum of terms of the form α^{p}2^{−nk_{o}} η̃(2^{−n}t)|t|^{2iα}Ψ(s_{i}t) where η̃ ∈ C_{c}^{∞}(R), supp(η̃) ⊂ supp(η) and Ψ is a Schwartz function which is 0(t^{2}) as t → 0 (note that Φ′(0) = ∫ xϕ(x)dx = 0 since ϕ is even). The worst power p is k_{o} which occurs when all derivatives land on the factor |t|^{2iα}.

Without loss of generality, let us suppose that G^{(k_{o})}(t) = α^{k_{o}}2^{−nk_{o}}η(2^{−n}t)|t|^{2iα}Ψ(s_{i}t). ¿From the above integral representation of F_{n,j}^{α}(t), we see that the main contribution to (1 + 2^{−2n}t^{2})^{m}|F_{n,j}^{α}(t)| occurs when |t| ∼ 2^{n} and in this case,

|F_{n,j}^{α}(t)| ≤ C|α|^{k_{o}}2^{(n−j)k_{o}}2^{−nk_{o}} min(1, (s_{i}2^{n})^{2}) ≤ C|α|^{k_{o}}2^{−jk_{o}} min(1, (2^{n}r_{i}θ)^{2}).

However we may write

F_{n,j}^{α}(t) = −2^{(n−j)k_{o}}2^{n−j} ∫_{0}^{1} ∫_{R} G_{n}^{(k_{o}+1)}(t − σ2^{j−n}s)sψ̂(s) ds dσ

and therefore we also have

|F_{n,j}^{α}(t)| ≤ C|α|^{k_{o}+1}2^{(n−j)k_{o}}2^{n−j}2^{−n(k_{o}+1)} min(1, (s_{i}2^{n})^{2}) ≤ C|α|^{k_{o}}2^{−jk_{o}}|α|2^{−j} min(1, (2^{n}r_{i}θ)^{2}),

establishing the claim.

Remarks. Theorem 1 holds also for Laplace-Beltrami operators on compact manifolds of dimension d. The proof is essentially the same as the proof of Theorem 1.

The hypotheses of Theorem 2 are satisfied for Laplace operators on Lie groups of polynomial growth. However, if L is a sub-Laplacian on the three dimensional Heisenberg group, then d = 4 but

C_{1}(1 + |α|)^{3/2} ≤ ∥L^{iα}∥_{L^{1}→L^{1,∞}} ≤ C_{ϵ}(1 + |α|)^{3/2+ϵ}.

See [16]. (See also [12]). The same estimates hold for a sub-Laplacian on SU(2) for which d = 4 and D = 0 (see [7]). Thus there are situations where the upper bound is better than the one given by Theorem 2 and where the lower-bound in Theorem 1 is false. For general groups of polynomial growth Theorem 2 gives the best known estimates, however as the above examples show, these bounds are not always best possible.

References

[1] G. Alexopoulos, Spectral multipliers on Lie groups of polynomial growth, Proc. Amer. Math. Soc. (3) 120 (1994), 973-979 .

[2] M. Christ, L^{p} bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc. (1) 328 (1991), 73-81.

[3] M. Christ, Lectures on singular integral operators, CBMS Regional Conference Series in Mathematics, 77, Amer. Math. Soc. Providence (1990).

[4] M. Christ, E.M. Stein, A remark on singular Calderón - Zygmund theory, Proc. Amer. Math. Soc. (3) 99 (1987), 71-75.

[5] M. Cowling, S. Meda, Harmonic analysis and ultracontractivity, Trans. Amer. Math. Soc. (2) 340 (1993), 733–752.

[6] M. Cowling, I. Doust, A. McIntosh, A. Yagi, Banach space operators with bounded H^{∞} functional calculus, J. Austral. Math. Soc. (Series A) 60 (1996), 51-89.

[7] M. Cowling, A. Sikora, Spectral multipliers for a sub-Laplacian on SU(2), in preparation.

[8] E.B. Davies, Heat kernels and spectral theory, Cambridge Univ. Press (1989).

[9] G. Dore, A. Venni, On the closedness of the sum of two closed operators, Math. Z. (2) 196 (1987), 189–201.

[10] A. Erdlyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher transcendental functions. Vol. I. Mc Graw-Hill (1953).

[11] C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9-36.

[12] W. Hebisch, Multiplier theorem on generalized Heisenberg groups, Colloq. Math. 65 (1993), 231–239.

[13] L. Hörmander, Estimates for translation invariant operators in L^{p} spaces, Acta Math. 104 (1960), 93-139.

[14] L. Hörmander, The analysis of linear partial differential operators III, Springer-Verlag (1983).

[15] S. Meda, A general multiplier theorem, Proc. Amer. Math. Soc. (3) 110 (1990), 639–647.

[16] D. Müller, E.M. Stein, On spectral multipliers for Heisenberg and related groups, J.Math. Pures Appl. 73 (1994), 413–440.

[17] J. Prüss, H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z. (3) 203 (1990), 429–452.

[18] A. Sikora, Sharp pointwise estimates on heat kernels, Quart. J. Math. Oxford (2) 47 (1996), 371-382.

[19] A. Sikora, On-diagonal estimates on Schrödinger semigroup kernels and reduced heat kernels, Comm. Math. Phys. 188 (1997), 233–249.

[20] E.M. Stein, Topics in harmonic analysis, Princeton University Press (1970).

[21] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press (1970).

[22] E.M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, (1993).

[23] M. Taylor, Partial differential equations, II, Springer-Verlag, New York (1996).

Adam Sikora, Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia (or University of Wrocław, KBN 2 P03A 058 14)

  E-mail address: sikora@maths.anu.edu.au

James Wright, School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia

  E-mail address: jimw@maths.unsw.edu.au