Functional calculus for slowly decaying kernels

by Waldemar Hebisch (Wrocław)^{1}

1. Introduction.

Let Ω be a metric space with the metric d and a Borel measure μ. Let

B(x, r) = {y ∈ Ω : d(x, y) < r}.

We assume that there are constants C, q (volume growth constants of Ω) such that

μ(B(x, sr)) ≤ Cμ(B(x, r))s^{q}

for all s > 1. The existence of such constants is equivalent to the doubling condition, but here we are interested in getting q as small as possible (at the cost of enlarging C).

The purpose of the paper is to present a reasonably general approach to functional calculus, multipliers and almost everywhere convergence theorems on Ω. We consider integral operators with kernels decaying polynomialy away from diagonal. Our methods are based on L^{2} estimates obtained from spectral theorem and careful use of weight functions.

The main question is to find sufficient conditions on positive definite operator A on Ω and on function F such that some of the following holds:

F (A) is bounded on L^{1}

F (A) is bounded on L^{p}, 1 < p < ∞ and of weak type (1,1). F^{∗}(A)f = sup_{t>0} |F (tA)f| is bounded on L^{p}, 1 < p < ∞ and of weak type (1,1).

The subject has long history. Let us only mention works [15], [5], [17], [21], [18]. Our topic is closely related to works on Riesz means, for example [16], [23], [3].

In a few classical examples our results are weaker then those previously known. However, it seems that proofs of, say, multiplier theorems go as follows. One splits the multiplier into diadically supported pieces. For each piece one gets estimates for dominant term via some kind of Plancherel formula, which requires a lot of specific knowledge - unavailable in our setting. Then there is error term estimate - in many cases long and very technical. Final step is use of covering or decomposition arguments to get estimates for the whole

1 Supported by KBN grant 2 P301 052 07. multiplier. Our methods seem well suited to handle error term estimates while for the main term we use essentially a trivial estimate. However, if better estimates for main term are known, we can use them. It is quite possible that in our general setting the trivial estimate is the best one. We also give an improved way of gluing estimates for pieces to get full multiplier theorem — unlike the case of singular integrals we need no smoothness assumptions (cancelations are provided by the L^{2} theory).

The main factor affecting our estimates is the volume growth rate. When the work on this paper begun this was believed to be the correct factor. Recent works [11], [20], [12],

[14] shows that the picture is much more complicated. 2.Banach Algebras

Definition: A continuous function φ : Ω × Ω → R is called submultiplicative if for all x,y,z

φ(x, y) ≥ 1

and

φ(x, y)φ(y, z) ≥ φ(x, z).

Of course if a,b > 0, then ω_{a} = (1 + d)^{a}, e^{bd}, ω_{a}e^{bd} are submultiplicative. Our functional calculus is based on Banach ∗-algebras whose elements are kernels K, K(x, y) being a complex number. For a submultiplicative function φ we write

∥K∥_{B(φ)} = sup_{y} ∫ |K(x, y)|φ(y, x)dμ(x)

K^{∗}(x, y) = K̄(y, x)

|K|_{B(φ)} = max{∥K∥_{B(φ)}, ∥K^{∗}∥_{B(φ)}}

and we define the Banach ∗-algebra with unit element by

B(φ) = {K : |K|_{B(φ)} < +∞} + CI.

The multiplication is defined by

K_{1}K_{2}(x, y) = ∫ K_{1}(x, s)K_{2}(s, y)dμ(s).

Obviously

∫ |K_{1}K_{2}(x, y)|φ(x, y)dμ(x)
≤ ∫ ∫ |K_{1}(x, s)K_{2}(s, y)|dμ(s)φ(x, y)dμ(x)

≤ ∫ ∫ |K_{1}(x, s)|φ(x, s)|K_{2}(s, y)|φ(s, y)dμ(s)dμ(x)
≤ ∫ ∥K_{1}∥_{B(φ)}|K_{2}(s, y)|φ(s, y)dμ(s)
≤ ∥K_{1}∥_{B(φ)}∥K_{2}∥_{B(φ)}

for every x. Hence

∥K_{1}K_{2}∥_{B(φ)} ≤ ∥K_{1}∥_{B(φ)}∥K_{2}∥_{B(φ)}

and

|K_{1}K_{2}|_{B(φ)} ≤ |K_{1}|_{B(φ)}|K_{2}|_{B(φ)}.

For a given kernel K we are going to estimate the norm of the element Ke^{inK} in B(ω_{a}). We will use the abbreviations

∥K∥_{a} = ∥K∥_{B(ω_{a})},
|K|_{a} = |K|_{B(ω_{a})}.

(2.1). Theorem. Let a > b ≥ 0 and Ω be as above. For every L there exists constant M depending only on a, b, L and volume growth constants C, q but independent of Ω, d, μ, K such that if

K = K^{∗},
∥K∥_{a} ≤ L,

sup_{x} μ(B(x, 1)) ∫ |K(x, y)|^{2}dμ(y) ≤ 1,

then for every n

|e^{inK}|_{b} ≤ M(1 + |n|)^{κ}

where κ = 2^{[(b+q/2)/(a−b)]}((b + q/2)(1 + 1/(a − b)) + 1).

Proof : Put A = exp(iK), then ∥A∥_{a} = |A|_{a} ≤ exp(L). Let p = q/2 + b, δ = a − b, r ≥ (n∥A∥_{a})^{1/δ} (r will be chosen later). Then |A|_{a}r^{−δ} ≤ 1/n.

We put

A_{0}(x, y) = \{ \begin{matrix}A(x, y) for d(x, y) < er \\ 0 otherwise, \end{matrix}

and

A_{k}(x, y) = \{ \begin{matrix}A(x, y) for re^{k} ≤ d(x, y) < re^{k+1} \\ 0 otherwise. \end{matrix}

If k ≥ 1,

∥A_{k}∥_{b} ≤ (re^{k})^{b−a}∥A∥_{a} = C_{1}e^{−kδ}

where C_{1} = r^{−δ}∥A∥_{a}. Also

∥A_{0}∥_{L^{2}→L^{2}} ≤ 1 + ∥A − A_{0}∥_{L^{2}→L^{2}} ≤ 1 + r^{−δ}|A|_{a} = C_{0} ≤ 1 + 1/n.

(2.2). Lemma. Let A_{i} be as above, α be a multiindex, E a kernel such that

sup_{y} μ(B(y, 1))∥E(·, y)∥_{L^{2}}^{2} ≤ M^{2}

and E(x, y) = 0 for d(x, y) > r_{0}. Then

∥ ∏_{i=1}^{|α|} A_{α(i)}E∥_{b} ≤ MC(r_{0} + r ∑ e^{α(i)})^{p} exp(−δ ∑ α(i))C_{1}^{|{i:α(i)>0}|}C_{0}^{|α|}

and

| ∏_{i=1}^{|α|} A_{α(i)}|_{b} ≤ C|α| (r ∑ e^{α(i)})^{p} exp(−δ ∑ α(i))C_{1}^{|{i:α(i)>0}|}C_{0}^{|α|}.

Proof :

∥ ∏_{i=1}^{|α|} A_{α(i)}∥_{L^{2}→L^{2}} ≤ ∏_{i=1}^{|α|} ∥A_{α(i)}∥_{L^{2}→L^{2}} ≤ exp(−δ ∑ α(i))C_{1}^{|{i:α(i)>0}|}C_{0}^{|α|}

Fix y. Put U = {x : d(x, y) ≤ r_{0} + er ∑ e^{α(i)}}. We have

∥ ∏_{i=1}^{|α|} A_{α(i)}E(·, y)(1 + d(·, y))^{b}∥_{L^{1}}

= ∥χ_{U} ∏_{i=1}^{|α|} A_{α(i)}E(·, y)(1 + d(·, y))^{b}∥_{L^{1}}

≤ sup_{x∈U}(1 + d(x, y))^{b}∥χ_{U}∥_{L^{2}}∥ ∏_{i=1}^{|α|} A_{α(i)}E(·, y)∥_{L^{2}}

≤ (1 + r_{0} + er ∑ e^{α(i)})^{b}|B(r_{0} + er ∑ e^{α(i)}, y)|^{1/2}

×∥ ∏_{i=1}^{|α|} A_{α(i)}∥_{L^{2}→L^{2}}∥E(·, y)∥_{L^{2}}

≤ C(r_{0} + r ∑ e^{α(i)})^{p} exp(−δ ∑ α(i))C_{1}^{|{i:α(i)>0}|}C_{0}^{|α|}M

which gives the first assertion. We prove the second assertion inductively. If α(|α|) > 0,

∥ ∏_{i=1}^{|α|} A_{α(i)}∥_{b} ≤ ∥ ∏_{i=1}^{|α|−1} A_{α(i)}∥_{b}∥A_{α(|α|)}∥_{b}

and we estimate the first factor by the inductive assumption. If α(|α|) = 0,

A_{0} = I + A′_{0}

where

sup_{y} μ(B(y, 1))∥A′_{0}(·, y)∥_{L^{2}} ≤ sup_{y} μ(B(y, 1))∥(A − I)(·, y)∥_{L^{2}} ≤ C

so

∥ ∏_{i=1}^{|α|−1} A_{α(i)}A_{0}∥_{b} ≤ ∥ ∏_{i=1}^{|α|−1} A_{α(i)}∥_{b} + ∥ ∏_{i=1}^{|α|−1} A_{α(i)}A′_{0}∥_{b}

and we estimate the first term by the inductive assumption and the second by the first assertion of the lemma. Applying ^{∗} to all the operators in the proof we get not only estimate for ∥ · ∥_{b} but also for | · |_{b}.

Let us note the following obvious conseqence of (2.2) .

(2.3). Lemma. If |α| ≤ n, then

| ∏_{i=1}^{|α|} A_{α(i)}|_{b} ≤ C|α|^{p+1} (re^{max α(i)})^{p} exp(−δ ∑ α(i))C_{1}^{|{i:α(i)>0}|}.

(2.4). Lemma. For every ε > 0 there exists constant C such that for every b, Ω, f : N ↦ R, n ∈ N and kernel A on Ω if

∥ ∏_{i=1}^{n} A_{α(i)}∥_{b} ≤ f(n) exp(−ε ∑ α(i))(1/n)^{|{i:α(i)>0}|}

then

|A^{n}|_{b} ≤ Cf(n).

Proof :

|A^{n}|_{b} = ∥A^{n}∥_{b} ≤ ∑_{|α|=n} ∥ ∏_{i=1}^{n} A_{α(i)}∥_{b}

≤ f(n) ∑_{|α|=n} exp(−ε ∑ α(i))(1/n)^{|{i:α(i)>0}|}

≤ f(n)(1 + 1/n ∑_{k=1}^{∞} e^{−εk})^{n}

≤ f(n) exp(∑_{k=1}^{∞} e^{−εk}) = Cf(n).

We expect main part of our kernels to live in the distance of order nr to the diagonal. In this region we sometimes have another estimate, and the following lemma allows us to use it efficiently.

(2.5). Lemma. For every ε > 0 there exists constant C such that for every b, Ω, f : N ↦ R, n ∈ N and kernels A and E on Ω if nC_{1} ≤ 1, E(x, y) = 0 for d(x, y) > nr and

∥ ∏_{i=1}^{n} A_{α(i)}E∥_{b} ≤ f(n) exp(−ε ∑ α(i))C_{1}^{|{i:α(i)>0}|}

then

sup_{y∈Ω} ∫_{d(x,y)>4nr} |A^{n}E(x, y)|(1 + d(x, y))^{b}dμ(x) ≤ Cf(n)nC_{1}.

Proof : As 4nr > enr + nr we have

sup_{y∈Ω} ∫_{d(x,y)>4nr} |A^{n}E(x, y)|(1 + d(x, y))^{b}dμ(x) ≤ ∑_{{|α|=n \atop α≠0}} ∥ ∏_{i=1}^{n} A_{α(i)}E∥_{b}

≤ f(n) ∑_{{|α|=n \atop α≠0}} C_{1}^{|\{i:α(i)>0\}|} exp(−ε ∑ α(i))

≤ f(n)nC_{1} ∑_{|α|=n} exp(−ε ∑ α(i))(1/n)^{|{i:α(i)>0}|}

≤ Cf(n)nC_{1}.

(2.6). Lemma. Let m be a natural number. For every n and every sequence (a_{i})_{i=1}^{n} of nonnegative real numbers there exists subset I of [1, n] ∩ N such that |I| ≤ 2^{m−1} − 1 and for every subset J of [1, n] ∩ N such that |J| ≤ |I| + 1 and J ∩ I = ∅ we have

m ∑_{j∈J} a_{j} ≤ ∑_{i=1}^{n} a_{i}

Proof : Without any loss of generality we assume that the sequence is nonincreasing and long enough, otherwise we renumerate it and add zeros.

Next

∑ a_{i} ≥ ∑_{k=0}^{m−1} ∑_{2^{k}}^{2^{k+1}−1} a_{i}

so there exists k ≤ m − 1 such that

∑ a_{i} ≥ m ∑_{2^{k}}^{2^{k+1}−1} a_{i}.

We take I = [1, 2^{k} − 1] ∩ N. Of course the first assertion holds. Let J be as in the second assertion. Since (a_{i}) is nonincreasing

m ∑_{j∈J} a_{j} ≤ m ∑_{2^{k}}^{2^{k+1}−1} a_{i}

which ends the proof.

(2.7). Lemma. Let m be a positive natural number, A and Ω be as above. There exists a constant C such that for all n

∥ ∏_{i=1}^{n} A_{α(i)}∥_{b} ≤ Cr^{(2^{m−1})p}n^{(2^{m−1})(p+1)} exp((\frac{p}{m} − δ) ∑ α(i))C_{1}^{|\{i:α(i)>0\}|}

Proof : Fix α. Take a_{i} = α(i) in Lemma (2.6) and choose subset I according to the Lemma. We write l = |I| and I = {i_{j} : j = 1, …, l}. Put i_{0} = 0 and i_{l+1} = n + 1. By the Lemma l ≤ 2^{m−1} − 1 and

m ∑_{j=1}^{l+1} max_{i_{j−1}<i<i_{j}} α(i) ≤ ∑ α(i).

Also, by (2.3) ,

∥ ∏_{i=i_{j−1}+1}^{i_{j}−1} A_{α(i)}∥_{b}

≤ Cn^{p+1} (r exp( max_{i_{j−1}<i<i_{j}} α(i)))^{p} exp(−δ ∑_{i_{j−1}<i<i_{j}} α(i))C_{1}^{|{i:i_{j−1}<i<i_{j},α(i)>0}|}

Next

∥ ∏_{i=1}^{n} A_{α(i)}∥_{b} ≤ ∥ ∏_{i=1}^{i_{1}−1} A_{α(i)}∥_{b} ∏_{j=1}^{l} (∥A_{α(i_{j})}∥_{b} ∥ ∏_{i=i_{j}+1}^{i_{j+1}−1} A_{α(i)}∥_{b})

≤ C(n^{p+1}r^{p})^{2^{m−1}} exp(p ∑_{j=1}^{l+1} max_{i_{j−1}<i<i_{j}} α(i)) exp(−δ ∑_{i} α(i))C_{1}^{|{α(i)>0}|}.

and the lemma follows.

End of the proof of (2.1) . We take m = [p/δ] + 1 and r = (n∥A∥_{a})^{1/δ}. Then (2.1) holds by (2.7) and (2.4) .

(2.8). Theorem. Let a > b, K, Ω satisfy assumptions of (2.1) and

s > 2^{[(b+q/2)/(a−b)]}((b + q/2)(1 + 1/(a − b)) + 1) + 1/2.

There exists C such that for all f ∈ H(s)

|f(K)|_{b} ≤ C∥f∥_{H(s)}.

Proof : We may assume that ∥K∥_{L^{2}} ≤ 1 (if not we replace K by K/∥K∥_{L^{2}} and adjust f). Therefore sp K ⊂ [−1, 1] and f(K) does not depend on the values of f outside [−1, 1]. Putting h = ϕf where ϕ ∈ C_{c}^{∞}([−2, 2]) and ϕ = 1 on [−1, 1] we have f(K) = h(K). Next

h(K) = ∑ ĥ(n)e^{inK}

and

|h(K)|_{b} ≤ ∑ |ĥ(n)||e^{inK}|_{b} ≤ M ∑ |ĥ(n)|(1 + |n|)^{κ} ≤ ∑ |ĥ(n)|(1 + |n|)^{s}(1 + |n|)^{κ−s}

≤ M∥h∥_{H(s)}(∑(1 + |n|)^{2(κ−s)})^{1/2} ≤ C∥f∥_{H(s)}

where κ and M are as in (2.1) .

(2.9). Lemma. Let I and J be closed intervals such that I ⊂ Int J ⊂ J ⊂ (−π, π). Let B(s) be C(s) or H(s), f ∈ B(s_{0}), supp f ⊂ I, l > 0. There exist functions f_{j}, j = 0, 1, … satisfying the following conditions

f = ∑ f_{j}
supp f_{j} ⊂ J
|f̂_{j}(k)| ≤ C(s_{0}, I, J, l)2^{−s_{0}j}(1 + max(0, |k| − 2^{j}))^{−l−3}∥f∥_{B(s_{0})}
∥f_{j}∥_{B(0)} ≤ C(s_{0}, I, J, l)2^{−s_{0}j}∥f∥_{B(s_{0})}

where C(s_{0}, I, J, l) depend only on s_{0}, I, J, l.

We choose smooth functions φ, ψ such that supp φ ⊂ J, φ|_{I} = 1, ψ = 1 on [−\frac{1}{2}, \frac{1}{2}], supp ψ ⊂ [−1, 1] and we put

h_{j}(x) = \{ \begin{matrix}∑ ψ(k)f̂(k)e^{ikx} for j = 0, \\ ∑[ψ(2^{−j}k) − ψ(2^{−j+1}k)]f̂(k)e^{ikx} otherwise, \end{matrix}

f_{j} = φh_{j}

Third condition holds because

f̂_{j}(k) = ∑_{r} φ̂(r)ĥ_{j}(k − r)

and |φ̂(k)| ≤ C(1 + |k|)^{−l−4}.

Remark. (2.9) is valid for very general scales of Banach spaces. In particular it is valid for Besov and Tribel-Lizorkin spaces.

(2.10). Theorem. For every b ≥ 0, s > (b + q/2)(1 + 2/(a − b)), a > 2b + q/2 there exist C such that

|f(K)K|_{b} ≤ C∥f∥_{C(s)}.

Proof : Without any loss of generality we assume that supp f ⊂ (−π, π). We decompose f as in (2.9) (with l > s). Fix k. We put n = 2^{k+2}, r = (n^{2}∥A∥_{a})^{(1/(a−b))}, p = b + q/2,

E(x, y) = \{ \begin{matrix}K(x, y) for d(x, y) < nr \\ 0 otherwise. \end{matrix}

Observe that

∫_{d(x,y)>4nr} |f_{k}(K)K(x, y)|(1 + d(x, y))^{b}dμ(y)

≤ ∫_{d(x,y)>4nr} |f_{k}(K)E(x, y)|(1 + d(x, y))^{b}dμ(y)

+∥f_{k}(K)∥_{b} ∫_{d(x,y)>nr} |K(x, y)|(1 + d(x, y))^{b}dμ(y) = I_{1} + ∥f_{k}(K)∥_{b}I_{2}

As I_{2} ≤ Cn^{−2} and p < δ, we get estimate for the second term as in (2.8) . Also

I_{1} ≤ ∑_{|j|≤2^{k+2}} |f̂_{k}(j)| ∫_{d(x,y)>4nr} |A^{j}E(x, y)|(1 + d(x, y))^{b}dμ(y) + ∑_{|j|>2^{k+2}} |f̂_{k}(j)| ∥A^{j}∥_{b}∥K∥_{b}

and by (2.2) , (2.5) and (2.9) (note that C_{1} ≤ \frac{1}{n^{2}} ), this is

≤ Cn^{−s}(nr)^{p}

Next, putting U = {x : d(x, y) ≤ 4nr}, we have

∫_{U} |f_{k}(K)K(x, y)|(1 + d(x, y))^{b}dμ(x)

≤ ∥χ_{U}∥_{L^{2}} sup_{x∈U}(1 + d(x, y))^{b}∥f_{k}(K)K(·, y)∥_{L^{2}}

≤ ∥f_{k}∥_{L^{∞}} sup_{x∈U}(1 + d(x, y))^{b}∥χ_{U}∥_{L^{2}}∥K(·, y)∥_{L^{2}}

≤ Cn^{−s}(nr)^{p}.

Gathering the estimates above we get

∥f_{k}(K)K∥_{b} ≤ Cn^{−s}(nr)^{p} ≤ C′2^{−kε}

for some ε > 0, which of course implies our claim.

(2.11). Theorem. Let I and J be closed intervals such that I ⊂ Int J ⊂ J. Assume a > 2b+q/2. If there exist κ, γ, χ such that κ ≥ 1, b ≥ 0, s > (b+q/2)(1+κ/(a−b))+2−κ, s > (b + γ/2)(1 + κ/(a − b)), s > χ + b(1 + κ/(a − b)) and for every f such that supp f ⊂ J

∫_{d(x,y)<r} |f(K)|(x, y)dx ≤ M(r^{γ/2}∥f∥_{L^{2}} + ∥f∥_{H(χ)}),

then for every f such that supp f ⊂ I

∥f(K)K∥_{b} ≤ M′∥f∥_{H(s)}.

Proof :The proof is similar to the proof of (2.10) . We choose r = (n^{κ}∥A∥_{a})^{(1/(a−b))} to get C_{1} = n^{−κ}, also l in (2.9) is made large enough. Then we use the assumption to estimate integral over the set U:

∫_{U} |f_{k}(K)K(x, y)|(1 + d(x, y))^{b}dμ(x)

≤ (1 + 4(nr)^{b}) ∫_{U} |f_{k}(K)K(x, y)|dμ(x)

≤ M(1 + 4(nr)^{b})((nr)^{γ/2}∥xf_{k}∥_{L^{2}} + ∥xf_{k}∥_{H(χ)})

≤ C(nr)^{b+γ/2}n^{−s} + C_{1}(nr)^{b}∥f_{k}∥_{H(χ+b(1+κ/(a−b)))}^{χ/(χ+b(1+κ/(a−b)))}∥f_{k}∥_{L^{2}}^{b(1+κ/(a−b))/(χ+b(1+κ/(a−b)))} ≤ C(nr)^{b+γ/2}n^{−s} + C_{2}(nr)^{b}(n^{−s})^{b(1+κ/(a−b))/(χ+b(1+κ/(a−b)))} ≤ C_{3}2^{−kε}

3. Multiplier theorems.

Let A be a non-negative self-adjoint densely defined operator on L^{2}(M, μ). Let E be the spectral measure of A. By the spectral theorem, we write

Af = ∫ λdE(λ)f

and

e^{−tA}f = ∫ e^{−λ}dE(λ)f.

We assume that

e^{−tA}f(x) = ∫ e^{−tA}(x, y)f(y)dμ(y)

where the kernels e^{−tA}(x, y) satisfy the following estimates :

there exist positive numbers a, m, α and C such that for all t

sup_{y} ∫ |e^{−tA}(x, y)|(1 + t^{−1/m}d(x, y))^{a}dμ(x) ≤ C

sup_{y} μ(B(y, t^{1/m})) ∫ |e^{−tA}(x, y)|^{2}dμ(x) ≤ C

(3.1). Theorem. Assume that the conditions above are satisfied and that

sup_{y,z} ∫ |e^{−tA}(x, y) − e^{−tA}(x, z)|dμ(x) ≤ Ct^{−α/m}d(y, z)^{α}.

If F ∈ H(s)_{loc},

s > 2^{[q/(2a)]}((q/2)(1 + 1/a) + 1) + 1/2

and for a non-zero φ ∈ C_{c}^{∞}(R_{+})

sup_{t>0} ∥F_{t}φ∥_{H(s)} ≤ M

where

F_{t}(λ) = F (tλ),

then F (A) = ∫ F (λ)dE(λ) is of weak type (1,1) and bounded on L^{p}(M), 1 < p < ∞.

(3.2). Lemma. If for some constants R < 1, M, a > 0, α > 0, a family of kernels {K_{n}}_{n=0}^{∞} satisfies

∥K_{n}∥_{B((1+R^{n}d)^{a})} ≤ M
∫ |K_{n}(x, y) − K_{n}(x, z)|dμ(x) ≤ MR^{nα}d(y, z)^{α},

then for some C depending only of R, a, α

sup_{z,y} ∫_{d(x,y)>2d(y,z)} ∑_{n} |K_{n}(x, y) − K_{n}(x, z)|dμ(x) ≤ MC

Proof : Fix y and z. Let

S = {x : d(x, y) > 2d(y, z)}.

Now

∫_{S} |K_{n}(x, y) − K_{n}(x, z)|dμ(x)

≤ ∫_{S} |K_{n}(x, y)|dμ(x) + ∫_{S} |K_{n}(x, z)|dμ(x) ≤ 2MR^{−na}d(y, z)^{−a}.

Then

∑ ∫_{S} |K_{n}(x, y) − K_{n}(x, z)|dμ(x)

≤ ∑ min(MR^{nα}d(y, z)^{α}, 2MR^{−na}d(y, z)^{−a})
≤ 2M(1 − R^{a})^{−1} + M(1 − R^{α})^{−1} ≤ CM.

Proof of (3.1) : First note that if μ(M) = ∞ then second assumption about e^{−tA} implies that the spectral measure of A has no atom at 0. If μ(M) < ∞ then this assumption implies that the spectral projector corresponding to 0 is bounded on L^{1} ( beeing bounded from L^{1} to L^{2} ⊂ L^{1}) and hence by interpolation and duality on all L^{p}, 1 ≤ p ≤ ∞. In any case we need to handle only (strictly) positive part of the spectrum. Choose φ ∈ C_{c}^{∞}(R_{+}) such that ∑ φ(2^{mn}λ) = 1 for λ > 0. Let

F_{n}(λ) = φ(λ)F (2^{−mn}λ),
G_{n}(λ) = F_{n}(− log(λ)),
H_{n}(λ) = F_{n}(− log(λ))λ^{−1},
e_{n} = e^{−2^{mn}A},
K_{n} = G_{n}(e_{n}) = H_{n}(e_{n})e_{n}.

We have

F (A) = ∑ F_{n}(2^{mn}A) = ∑ K_{n}.

By the assumption,

sup_{y} ∫ |e_{n}(x, y)|(1 + 2^{−n}d(x, y))^{a}dμ(x) ≤ C

sup_{y} μ(B(y, 2^{n})) ∫ |e_{n}(x, y)|^{2}dμ(x) ≤ C

and of course,

∥G_{n}∥_{H(s)} ≤ C′

therefore replacing d by 2^{−n}d we may apply (2.8) to get

∥K_{n}∥_{B((1+2^{−n}d)^{ε})} ≤ M

for sufficiently small ε > 0. Moreover

|K_{n}(x, y) − K_{n}(x, z)| ≤ ∫ |H_{n}(e_{n})(x, s)||e_{n}(s, y) − e_{n}(s, z)|dμ(s)

hence

∫ |K_{n}(x, y) − K_{n}(x, z)|dμ(x) ≤ ∥H_{n}(e_{n})∥_{0} ∫ |e_{n}(s, y) − e_{n}(s, z)|dμ(s) ≤ M2^{−αn}d(y, z)^{α}

This means that assumptions of (3.2) are satisfied.

Now, we use the general theory of Calderón-Zygmund operators (see for example Coifman-Weiss [4]). (3.3). Theorem. If K = ∑ K_{k}, K is bounded on L^{p}, p > 1, K_{k}Ψ_{j} = 0 for k > j, K_{k}Ψ_{j} = K_{k} for k < j and

∫ |K_{k}(x, y)|(1 + 2^{−k}d(x, y))^{ϵ} ≤ C
|Ψ_{k}(x, y)| ≤ C(1 + 2^{−k}d(x, y))^{−q−ϵ}(B(x, 2^{k})^{−1} + B(y, 2^{k})^{−1})

then K is of weak type 1-1.

Put

ρ_{t}(x, y) = (1 + t^{−1}d(x, y))^{−q−ε}(μ(B(x, t))^{−1} + μ(B(y, t))^{−1}),

M̃f(x) = sup_{t>0} ∫ ρ_{t}(x, y)|f(y)|dy.

(3.4). Lemma.

M̃f(x) ≤ CMf(x).

In particular M̃ is bounded on L^{p}, for all 1 < p ≤ ∞. Proof :There exists C such that

μ(B(x, 2d(x, y))) ≤ C(1 + t^{−1}d(x, y))^{q}μ(B(x, t)),
μ(B(x, 2d(x, y))) ≤ μ(B(y, 4d(x, y))) ≤ C(1 + t^{−1}d(x, y))^{q}μ(B(y, t))

for all x, y, t > 0. Hence

ρ_{t}(x, y) ≤ C′(μ(B(x, max(t, 2d(x, y)))))^{−1}(1 + t^{−1}d(x, y))^{−ε}.

Fix t. We have

∫ ρ_{t}(x, y)|f(y)|dy ≤ C′′ ∑_{k=0}^{∞} 2^{−ε}μ(B(x, 2^{k}t))^{−1} ∫_{B(x,2^{k}t)} |f(y)|dy

≤ C′′′Mf(x).

If f is an arbitrary L^{1} function and λ is a positive real number, then either μ(M) ≤ ∥f∥_{L^{1}}/λ and

μ({x : |Kf(x)| > λ}) ≤ μ(M) ≤ ∥f∥_{L^{1}}/λ

or we apply to f the Calderon-Zygmund decomposition at height λ (see for example Coifman-Weiss [4], Chapitre 3, Theoreme (2.2) ) and obtain balls B(x_{i}, r_{i}) such that putting P = ⋃ B(x_{i}, r_{i}) we have

∥f|_{M−P} ∥_{L^{∞}} ≤ Cλ

∫_{B(x_{i},r_{i})} |f| ≤ Cλμ(B(x_{i}, r_{i}))

∑ μ(B(x_{i}, r_{i})) ≤ C∥f∥_{L^{1}}/λ

and every x ∈ M belongs to at most C different B(x_{i}, r_{i}).

Put Q_{i} = B(x_{i}, r_{i}), Q_{i}^{∗} = B(x_{i}, 2r_{i}), S_{i} = Q_{i} − ∪_{j<i}Q_{j}, f_{i} = χ_{S_{i}}f, k_{i} = [log_{2}(r_{i})].

(3.5). Lemma. There exists C such that for all i

∑_{n} ∫_{(Q_{i}^{∗})^{c}} |K_{n}(1 − Ψ_{k_{i}})f_{i}|(x)dx ≤ C∥f_{i}∥_{L^{1}}.

Proof :

∫_{(Q_{i}^{∗})^{c}} |K_{n}f_{i}|(x)dx ≤ ∥f_{i}∥_{L^{1}} sup_{y∈Q_{i}} ∫_{(Q_{i}^{∗})^{c}} |K_{n}(x, y)|dx

≤ ∥f_{i}∥_{L^{1}} sup_{y} ∫_{d(x,y)>2^{k_{i}−1}} |K_{n}(x, y)|dx

≤ 2^{ϵ(n−k_{i}+1)}∥f_{i}∥_{L^{1}} sup_{y} ∫_{d(x,y)>2^{k_{i}−1}} |K_{n}(x, y)|(1 + 2^{−n}d(x, y))^{ϵ}dx

≤ C2^{ϵ(n−k_{i})}∥f_{i}∥_{L^{1}}

By assumption, K_{n}(1 − Ψ_{k_{i}}) = 0 for n > k_{i} and K_{n}(1 − Ψ_{k_{i}}) = K_{n} for n < k_{i}. Hence

∑_{n} ∫_{(Q_{i}^{∗})^{c}} |K_{n}(1 − Ψ_{k_{i}})f_{i}|(x)dx ≤ ∥K_{k_{i}}(1 − Ψ_{k_{i}})f_{i}∥_{L^{1}} + ∑_{n<k_{i}} ∫_{(Q_{i}^{∗})^{c}} |K_{n}f_{i}|(x)dx

≤ C∥f_{i}∥_{L^{1}} ∑_{n≤k_{i}} 2^{ϵ(n−k_{i})}

≤ C∥f_{i}∥_{L^{1}}.

(3.6). Lemma. For every 1 ≤ p < ∞ there exist C_{p} such that

∥ ∑ Ψ_{k_{i}}f_{i}∥_{L^{p}}^{p} ≤ C_{p}λ^{p−1}∥f∥_{L^{1}}.

Proof :Put τ_{i} = ρ_{2^{k_{i}}}. It is easy to check (using doubling condition) that

sup_{y∈Q_{i}} τ_{i}(x, y) ≤ C inf_{y∈Q_{i}} τ_{i}(x, y)

with C uniform in x and i. Fix i and x and choose y_{0} ∈ Q_{i}.

|Ψ_{k_{i}}f_{i}| ≤ ∫ τ_{i}|f_{i}|(y)dμ(y)
≤ Cλμ(Q_{i})τ_{i}(x, y_{0})
≤ C′λτ_{i}χ_{Q_{i}}.

Let r = p/(p − 1). If h ∈ L^{r}, h ≥ 0, then

|(h, τ_{i}χ_{Q_{i}})| = |(τ_{i}h, χ_{Q_{i}})| ≤ (M̃h, χ_{Q_{i}}).

By (3.4) ,

(h, ∑ |Ψ_{k_{i}}f_{i}|) ≤ C(M̃h, ∑ λχ_{Q_{i}}) ≤ C∥h∥_{L^{r}}∥ ∑ λχ_{Q_{i}}∥_{L^{p}}

By properties of Calderón-Zygmund decomposition ∥ ∑ λχ_{Q_{i}}∥_{L^{p}}^{p} ≤ C ∑ λ^{p}μ(Q_{i}) ≤ C′λ^{p−1}∥f∥_{L^{1}}, which ends the proof of (3.6) .

Let

g = f − ∑_{i}(1 − Ψ_{k_{i}})f_{i} = f − ∑_{i} f_{i} + ∑_{i} Ψ_{k_{i}}f_{i}

so

Kf = K(∑_{i}(1 − Ψ_{k_{i}})f_{i} + g) = ∑_{n} ∑_{i} K_{n}(1 − Ψ_{k_{i}})f_{i} + Kg

We have

∥g∥_{L^{p}}^{p} ≤ p(∥f − ∑_{i} f_{i}∥_{L^{p}}^{p} + ∥ ∑_{i} Ψ_{k_{i}}f_{i}∥_{L^{p}}^{p})

≤ p(∥f − ∑_{i} f_{i}∥_{L^{∞}}^{p−1}∥f − ∑_{i} f_{i}∥_{L^{1}} + Cλ^{p−1}∥f∥_{L^{1}}) ≤ C′λ^{p−1}∥f∥_{L^{1}}.

Put E = ∪_{i}Q_{i}^{∗}. By (3.5) ,

∫_{M−E} | ∑_{n} ∑_{i} K_{n}(1 − Ψ_{k_{i}})f_{i}|(x)dx ≤ ∑_{n} ∑_{i} ∫_{(Q_{i}^{∗})^{c}} |K_{n}(1 − Ψ_{k_{i}})f_{i}|(y)dy

≤ ∑_{i} C∥f_{i}∥_{L^{1}} ≤ C′∥f∥_{L^{1}}.

Finally

|{|Kf| > λ}| ≤ |{|Kg| > λ/2}| + |{| ∑_{n} ∑_{i} K_{n}(1 − Ψ_{k_{i}})f_{i}| > λ/2}|

≤ \frac{(2∥Kg∥_{L^{p}})^{p}}{λ^{p}} + |E| + \frac{2 ∫_{M−E} | ∑_{n} ∑_{i} K_{n}(1 − Ψ_{k_{i}})f_{i}|(x)dx}{λ}

≤ C(\frac{λ^{p−1}∥f∥_{L^{1}}}{λ^{p}} + \frac{∥f∥_{L^{1}}}{λ}) ≤ 2Cλ^{−1}∥f∥_{L^{1}}

which ends the proof of (3.3) .

(3.7). Theorem. If a > q, l > (1 + 2/a)q/2 + 1, p > 1 and for a non-zero φ ∈ C_{c}^{∞}(R_{+}),

∫_{0}^{∞}t^{−1}∥φδ(t)F ∥_{W_{l}^{p}}dt < ∞,

then

F^{∗}f = sup_{t>0} |δ(t)F (A)f|

is of weak type (1,1) and bounded on L^{p} for 1 < p ≤ ∞.

Remark. The index l in (3.7) can not be essentially lowered. This contrasts with homogeneous multipliers on R^{n}, (see [22]) where one can take l close to (n + 1)/2 (or use derivatives in L^{2} and l close to n/2). To see this let A be operator on C_{c}^{∞}(R^{2}) defined by the formula

Âf = ψf̂

where ˆ denotes the Fourier transform. Assume that ψ is homogeneous with respect to anisotropic dilations, that is ψ(tx, t^{2}y) = t^{k}ψ(x, y) and that the level set {(x, y) : ψ(x, y) = 1} contains an interval I not parallel to any of the coordinate axes. Let supp f̂ be contained in small neighbourhood of a point x in I. Take F (λ) = (1 − λ)^{(1−ϵ)}. Then

(F (A)f)(x, y) ≈ (x^{2} + y^{2})^{−1+ϵ/2}

when (x, y) lies in a bounded distance from direction normal to I and F (A)f is small outside this set. If we take F (tA)f then we get similar estimate but the set where F (tA)f is large changes. More precise, it is obtained applying anisotropic dilation to the set where F (A)f is large. Anisotropic dilations act transitively on directions different from axes, so

F^{∗}(A)f(x, y) ≈ (x^{2} + y^{2})^{−1+ϵ/2}

on a cone with nonempty interior. But then clearly F^{∗}(A) is not of weak type 1-1. If k is large then A satisfies our estimates with large a. One can modify this example to get differential A, (then level set must be tangent of high order to I).

(3.8). Theorem. If a > q, s > (1 + 2/a)q/2 and for a non-zero φ ∈ C_{c}^{∞}(R_{+}) and a constant C

sup_{t>0} ∥φδ(t)F ∥_{C(s)} ≤ C,

then F (A) is of weak type (1, 1) and bounded on L^{p} for 1 < p < ∞.

Remark. With other assumptions as in (3.8) , if

∫_{0}^{∞}t^{−1}∥φδ(t)F ∥_{C(s)}dt < ∞,

then F (A) is bounded on L^{1}.

Remark. (3.8) applied to the sublaplacean on Lie group of polynomial growth gives result essentially equivalent to that of G. Aleksopoulos [1]. (3.9). Lemma. There exists C such that for all x, y and t

|e^{tA}(x, y)| ≤ Cμ(B(y, t^{1/m}))^{−1}.

Proof. We are going to prove that for all ε > q/2 there exists C such that

|e^{tA}(x, y)| ≤ C(1 + t^{−1/m}d(x, y))^{−(a−ε)}(μ(B(x, t^{1/m}))μ(B(y, t^{1/m})))^{−1/2} for all x, y and t. This easily implies our claim.

The estimate above is trivially true for ε = a. We have (using Schwartz inequality) |e^{2tA}(x, y)|(1 + t^{−1/m}d(x, y))^{(a−ε)}(μ(B(x, t^{1/m}))μ(B(y, t^{1/m})))^{1/2} = ∫ |e^{tA}(x, s)e^{tA}(s, y)|(μ(B(x, t^{1/m}))μ(B(y, t^{1/m})))^{1/2}(1 + t^{−1/m}d(x, y))^{(a−ε)}ds

≤ sup_{y} μ(B(y, t^{1/m})) ∫ |e^{tA}(s, y)|^{2}(1 + t^{−1/m}d(s, y))^{2(a−ε)}ds

≤ sup_{y} ∫ |e^{tA}(s, y)|(1 + t^{−1/m}d(s, y))^{a}ds

× sup_{s,y} μ(B(y, t^{1/m}))|e^{tA}(s, y)|(1 + t^{−1/m}d(x, y))^{(a−2ε)}

≤ C sup_{s,y}(μ(B(x, t^{1/m}))μ(B(y, t^{1/m})))^{1/2}|e^{tA}(s, y)|(1 + t^{−1/m}d(x, y))^{(a−2ε+q/2)}

Repeating this we can get ε − q/2 arbitrarily small and thus (3.9) is proved.

We fix φ and ψ such that φ, ψ are in C^{∞}(R), supp φ ⊂ [2^{−m}, 2^{m/2}], (∀x > 0) ∑ φ(2^{mk}x) = 1, and supp ψ ⊂ [−1, 1], with ψ(x) = 1 for x ∈ [0, 2^{−m/2}]. Let

φ_{k}(λ) = φ(2^{mk}λ),

ψ_{k}(λ) = ψ(2^{mk}λ),
Ψ_{k} = ψ_{k}(A),
F_{k}(λ) = φ_{k}(λ)F (λ),
G_{k}(λ) = F_{k}(−2^{−mk} log(λ))λ^{−1},
e_{k} = e^{−2^{mk}A},
W_{k} = ψ_{k}(−2^{−mk} log(λ))λ^{−2},
K_{k} = G_{k}(e_{k})e_{k}.

We have

F (A) = ∑ F_{k}(A) = ∑ G_{k}(e_{k})e_{k} = ∑ K_{k}.

By the assumption,

sup_{y} ∫ |e_{k}(x, y)|(1 + 2^{−k}d(x, y))^{a}dμ(x) ≤ C

sup_{y} μ(B(y, 2^{k})) ∫ |e_{k}(x, y)|^{2}dμ(x) ≤ C

and of course,

∥G_{k}∥_{C(l)} ≤ C′

therefore replacing d by 2^{−k}d we may apply (2.10) to get

∥K_{k}∥_{B((1+2^{−k}d)^{ε})} ≤ M

for sufficiently small ε > 0.

Also

ψ_{k}(A) = W_{k}(e_{k})e_{k}^{2}

and for any s

∥W_{k}∥_{H(s)} ≤ C_{s}

therefore replacing d by 2^{−k}d we may apply (2.8) to get

∫ |W_{k}(e_{k})|(x, y)(1 + 2^{−k}d(x, y))^{a−ϵ} ≤ C

By (3.9) (and symmetry) we get

sup_{y} |W_{k}(e_{k})e_{k}|(x, y) ≤ μ(B(x, 2^{k}))^{−1}

Next

|ψ_{k}(A)|(x, y) ≤ ∫ |W_{k}(e_{k})e_{k}|(x, s)|e_{k}|(s, y)ds

≤  ∫_{d(x,s)≥d(x,y)/2} |W_{k}(e_{k})e_{k}|(x, s)|e_{k}|(s, y)ds +  ∫_{d(y,s)≥d(x,y)/2} |W_{k}(e_{k})e_{k}|(x, s)|e_{k}|(s, y)ds

≤ sup_{s} |e_{k}|(s, y) ∫_{d(x,s)≥d(x,y)/2} |W_{k}(e_{k})e_{k}|(x, s)ds

+ sup_{s} |W_{k}(e_{k})e_{k}|(x, s) ∫_{d(y,s)≥d(x,y)/2} |e_{k}|(s, y)ds

≤ C(1 + t^{−1/m}d(x, y))^{−(a−ε)}(μ(B(x, t^{1/m}))^{−1} + μ(B(y, t^{1/m}))^{−1})

In other words

|ψ_{k}(A)|(x, y) ≤ C(1 + 2^{−k}d(x, y))^{−(a−ε)}(μ(B(x, 2^{−k}))^{−1} + μ(B(y, 2^{−k}))^{−1}) which is the second assumption in (3.3) . This ends the proof of (3.8) .

The following lemma is all what is needed to end the proof of (3.7) . (3.10). Lemma.

∫ sup_{t>0} |F (tA)φ_{k}(A)|(x, y)(1 + 2^{−k}d(x, y))^{ϵ}dx ≤ C ∫_{0}^{∞}t^{−1}∥φδ(t)F ∥_{W_{l}^{p}}dt < ∞.

Proof :We have

∫ sup_{t>0} |F (tA)φ_{k}(A)|(x, y)(1 + 2^{−k}d(x, y))^{ϵ}dx

≤ ∑_{j} ∫ sup_{t>0} |F_{j}(tA)φ_{k}(A)|(x, y)(1 + 2^{−k}d(x, y))^{ϵ}dx

and

∑_{j} ∥δ(2^{−mj})F_{j}∥_{W_{l}^{p}} ≤ C ∫_{0}^{∞}t^{−1}∥φδ(t)F ∥_{W_{l}^{p}}dt

so it is enough to show that

∫ sup_{t>0} |F_{j}(tA)φ_{k}(A)|(x, y)(1 + 2^{−k}d(x, y))^{ϵ}dx ≤ C∥δ(2^{−mj})F_{j}∥_{W_{l}^{p}}.

Without any loss of generality we may assume that k = 0 and j = 0. Indeed, otherwise we replace d by 2^{−k}d and F by δ(2^{−mj})F. We write

e_{n} = exp(ine^{−A})e^{−A},

H_{t}(λ) = (φ_{0})(− log(λ))F_{0}(−t log(λ))/λ.

We have

F_{0}(tA)φ(A) = H_{t}(e^{−A})e^{−A} = ∑_{n} Ĥ_{t}(n)e_{n}

and H_{t} = 0 for t ∉ [2^{−2m}, 2^{2m}]. It follows

sup_{t>0} |φ(A)F_{0}(tA)|(x, y) ≤ ∑_{n} sup_{2^{2m}≥t>2^{−2m}} |Ĥ_{t}(k)||e_{n}(x, y)|

and

∫ sup_{t>0} |F_{0}(tA)φ(A)|(x, y)(1 + 2^{−k}d(x, y))^{ϵ}dx

≤ ∑_{n} sup_{2^{2m}≥t>2^{−2m}} |Ĥ_{t}(n)| ∫ |e_{n}(x, y)|(1 + 2^{−k}d(x, y))^{ϵ}dx

≤ C∥F_{0}∥_{W_{l}^{p}} ∑_{n}(1 + |n|)^{−l}∥e_{n}∥_{ϵ}

≤ C′∥F_{0}∥_{W_{l}^{p}} ∑_{n}(1 + |n|)^{−l}(1 + |n|)^{(1+2/(a−ϵ))q/2} ≤ C′′∥F_{0}∥_{W_{l}^{p}}.

Let M be a homogeneous group (cf. [7]) of homogeneous dimension Q, that is a Lie group equipped with one-parameter family {δ_{t}}_{t>0} of automorphisms such that for all x ∈ M

lim_{t→0} δ_{t}x = e

where e is the neutral element of M and for every compact set A ⊂ M

μ(δ_{t}(A)) = t^{Q}μ(A)

where μ is the Haar measure. Such a Q is not unique — as additional normalization we require all eigenvalues of D_{e}δ_{t} to be at most t for t < 1 and to have t as one of the eigenvalues. We assume that A is a left-invariant hypoelliptic differential operator, homogeneous of degree m > 0, that is (for f in the domain of A)

A(f ∘ δ_{t}) = t^{m}(Af) ∘ δ_{t}.

Such an A is usually called Rockland operator. Since A is left-invariant we have

F (A)f = f ∗ H_{F}

where H_{F} is a distribution on M called the kernel of F (A). In our setting we have a kind of Plancherel formula:

∥H_{F} ∥_{L^{2}} = c ∫ |F (A)|^{2}(x)x^{−1+Q/m}dx.

To see this note that as ∥H_{F} ∥_{L^{2}} = H_{|F |^{2}}(e) the formula above is equivalent to

H_{F} (e) = c ∫ F (A)(x)x^{−1+Q/m}dx.

For F (x) = e^{−x} both sides are finite (and nonzero) so one can choose c to have equality. Then homogeneity shows that the set of F for which equality holds is closed under dilations. Thus the equality holds for linear combinations of exponentials - that is on a set dense in L^{1}(x^{−1+Q/m}dx), which shows the formula. We fix a riemmanian metric d on G. Note that it can happen that q (which describes growth of balls for riemmanian metric) is smaller then Q.

(3.11). Theorem. If A and G are as above, s > q/2 and for a non-zero φ ∈ C_{c}^{∞}(R_{+}) and a constant C

sup_{t>0} ∥φδ(t)F ∥_{H(s)} ≤ C,

then F (A) is of weak type (1, 1) and bounded on L^{p} for 1 < p < ∞.

Remark. If A is a homogeneous sublaplacean this theorem reduces to the theorem of M. Christ [2] and G. Mauceri and S. Meda [19]. If A is nondifferential then possible values of a are bounded and we must assume that a > q and s > (1 + 2/a)q/2. J. Dziubański [6] showed how to apply results like (3.11) for some nondifferential A when a is small to get s = q/2 + ϵ with arbitrarily small ϵ > 0.

Proof :First choose a so that a > Q (hence a > q), s > (1 + 2/a)q/2. Then we proceed as in the proof of (3.8) . The difference is that instead of 2^{−k}d we use d_{k}(x, y) = d(δ(2^{−k})x, δ(2^{−k})y) where d is some fixed left-invariant riemmanian metric on G. Also thanks to the Plancherel formula we may apply (2.11) instead of (2.10) . Indeed, we should estimate ∥K_{k}∥_{B((1+d_{k})^{ε})} (for Ψ_{k} we directly re-use the proof of (3.8) ). Using dilations, we reduce to problem to the estimate for ∥K_{0}∥_{B((1+d)^{ε})}. We write f̃(x) = f(e^{−x}) so f(e_{0}) = f̃(A) (and we assume that f have support in some fixed interval I contained in positive reals — we want to be away from 0). Then

∫_{d(x,e)<r} |f(e_{0})|(x, e)dx ≤ |{x : d(x, e) < r}|^{1/2}∥f(e_{0})∥_{L^{2}}

≤ Cr^{q/2}∥f̃(A)∥_{L^{2}} = C_{1}r^{q/2}(∫ |f̃(x)|^{2}x^{−1+Q/m}dx)^{1/2}

≤ C_{2}r^{q/2}∥f̃∥_{L^{2}} ≤ C_{3}r^{q/2}∥f∥_{L^{2}}.

Since our kernels are left-invariant the estimate above means that assumptinons of (2.11) are satisfied with κ = 2, γ = q, χ = 0 so we get uniform bound on ∥K_{k}∥_{B((1+d_{k})^{ε})}.

To finish the proof choose a homogeneous (dilation invariant) and left-invariant metric ρ on G (see for example [13]). We have

(1 + ρ(x, y)) ≤ C(1 + d(x, y))

and because ρ is homogeneous

(1 + 2^{−k}ρ(x, y)) ≤ C(1 + d_{k}(x, y)).

Hence assumptions of (3.3) are satisfied with d replaced by ρ which ends the proof.

Suppose that M is a smooth compact manifold of dimension q without boundary. Assume that A is an elliptic differential operator on M of order m which is positive definite on L^{2} with respect to a smooth positive density dμ. Fix a riemmanian metric d on M.

(3.12). Lemma. If M and A are as above, q > 1, m is order of A, supp f ⊂ [1/4, 4], t < 1, then there is C such that

∥f(tA)∥_{L^{1},L^{2}}^{2} ≤ Ct^{−q/m}(t^{q/m}∥f∥_{H(q/2)}^{2} + ∥f∥_{L^{2}}^{2}).

Proof :We need Hörmander’s estimate on spectral projections (see [16]) for a > 1/4 :

∥E([a, a + a^{(m−1)/m}))∥_{L^{1},L^{2}}^{2} ≤ Ca^{(q−1)/m}.

Put a = t^{−1}/4, h = t^{−(m−1)/m)}, n = 15([t^{−1/m}] + 1). Then, using orthogonality and Hörmander’s estimate we get

∥f(tA)∥_{L^{1},L^{2}}^{2} ≤ ∑_{k=0}^{n} ∥f(tA)E([a + kh, a + (k + 1)h))∥_{L^{1},L^{2}}^{2}

≤ Ct^{−(q−1)/m} ∑_{k=0}^{n} sup_{x∈[(a+kh)t,(a+(k+1)ht)} |f(x)|^{2}.

For any interval I we have

sup_{x∈I} |f|^{2} ≤ C(|I|^{−1}∥f∥_{L^{2}(I)}^{2} + |I|∥∂_{x}f∥_{L^{2}(I)}^{2})

so

∑_{k=0}^{n} sup_{x∈[(a+kh)t,(a+(k+1)ht)} |f(x)|^{2} ≤ C((ht)^{−1}∥f∥_{L^{2}}^{2} + (ht)∥∂_{x}f∥_{L^{2}}^{2})

Also

∥∂_{x}f∥_{L^{2}}^{2} ≤ C∥f∥_{L^{2}}^{2(q−2)/q}∥f∥_{H(q/2)}^{4/q} ≤ C′((ht)^{−2}∥f∥_{L^{2}}^{2} + (ht)^{q−2}∥f∥_{H(q/2)}^{2}).

Finally

∥f(tA)∥_{L^{1},L^{2}}^{2} ≤ Ct^{−(q−1)/m}((ht)^{−1}∥f∥_{L^{2}}^{2} + (ht)^{q−1}∥f∥_{H(q/2)}^{2})
≤ Ct^{−q/m}(∥f∥_{L^{2}}^{2} + t^{q/m}∥f∥_{H(q/2)}^{2}).

(3.13). Theorem. If M is a compact riemmanian manifold, q is the dimension of M, A is an elliptic differential operator of order m, positive definite on L^{2} with respect to a smooth measure μ, F ∈ C(R), s > q/2 and for a non-zero φ ∈ C_{c}^{∞}(R_{+}) and a constant C

sup_{t>0} ∥φδ(t)F ∥_{H(s)} ≤ C,

then F (A) is of weak type (1, 1) and bounded on L^{p} for 1 < p < ∞.

Remark. A variant of (3.13) is valid on noncompact manifolds. One needs bounded geometry assumptions to have global bound in Hörmander’s estimate and to control local behaviour of the semigroup. Also conclusion asserts only boundedness of operator with kernel restricted to a neighbourhood of diagonal.

Remark. Theorem remains valid form pseudodifferential operators. However, the a one gets depend on m so one must pass to the powers of A (like in [6]).

Proof :We give the proof only for q > 1. (If q = 1 one must replace 1/m in (3.12) by a larger number, and then tediously check that the proofs remain valid).

From the local Sobolev lemma we deduce that if F ∈ C_{c}(R), then F (A) has bounded kernel, hence is bounded on all L^{p}, 1 ≤ p ≤ ∞. Therefore we can assume that supp F ⊂ [1, ∞). The semigroup generated by differential operator A satisfies our assumptions with any a > 0 as long as t is bounded (see for example [10]). Because of our assumption about support of F estimates for large t are not needed. We proceed as in the proof of (3.8) but using (3.12) and (2.11) instead of (2.10) . Indeed we should estimate ∥K_{k}∥_{B((1+d_{k})^{ε})} for k < 0. Fix k. Put d_{k} = 2^{−k}d. We write f̃(x) = f(e^{−x}) so f(e_{k}) = f(e^{−2^{mk}A}) = f̃(2^{mk}A). Then

∫_{d_{k}(x,e)<r} |f(e_{k})|(x, y)dx ≤ |{x : d(x, e) < r2^{k}}|^{1/2}∥f(e_{k})(·, y)∥_{L^{2}}

≤ Cr^{q/2}2^{kq/2}∥f̃(2^{mk}A)∥_{L^{1},L^{2}}
≤ C_{1}r^{q/2}2^{kq/2}2^{−kq/2}(2^{kq/2}∥f̃∥_{H(q/2)} + ∥f̃∥_{L^{2}})
≤ C_{2}r^{q/2}(2^{kq/2}∥f∥_{H(q/2)} + ∥f∥_{L^{2}})
≤ C_{3}(r^{q/2}∥f∥_{L^{2}} + ∥f∥_{H(q/2)}).

In the last line we use compactness of our manifold – we may assume that r2^{k} is bounded. Thus, we checked that assumptions of (2.11) are satisfied with κ = 2, γ = q, χ = q/2.

References

[1] G. Alexpoulos, Spectral multipliers on Lie groups of polynomial growth, Proc. AMS 120 (1994), 973-979.

[2] M. Christ, L^{p} bound for spectral multiplier on nilpotent groups, TAMS 328 (1991), 73-81.

[3] M. Christ, Ch. Sogge, The weak type L^{1} convergence of eigenfunction expansions for pseudodifferential operators, Invent. Math. 94 (1988), 421–453.

[4] R. Coifman, Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes, Springer LN 242,Berlin 1971.

[5] J. Dixmier, Opérateurs de rang fini dans les représentations unitaires, Publ. Math. IHES 6 (1960), p. 13–25.

[6] J. Dziubański, A remark on a Marcinkiewicz-Hörmander multiplier theorem for some non-differential convolution operators, Coll. Math. 58 (1989), 77–83.

[7] G. B. Folland, E. M. Stein, Hardy spaces on Homogeneous Groups, Princeton University Press, 1982.

[8] P. Głowacki, The Rockland condition for nondifferential convolution operators, Duke Math. J. 58 (1989), 371–395.

[9] W. Hebisch, A multiplier theorem for Schrödinger operators, Coll. Math. 60/61 (1990), 659-664.

[10] W. Hebisch, Almost everywhere summability of eigenfunction expansions associated to elliptic operators, Studia Math. 96 (1990), 263-275.

[11] W. Hebisch, The subalgebra of L^{1}(AN) generated by the laplacean, Proc. AMS 117 (1993), 547-549.

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

[13] W. Hebisch, A. Sikora, A smooth subadditive homogeneous norm on a homogeneous group, 96 Studia Math. 96 (1990) 231–236.

[14] W. Hebisch, J. Zienkiewicz, Multiplier theorem on generalized Heisenberg groups II, Coll. Math. 69 (1995) 29–36.

[15] L. Hörmander, Estimates for translation invariant operators in L^{p} spaces, Acta Math. 104 (1960), 93–140.

[16] L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218.

[17] A. Hulanicki, Subalgebra of L_{1}(G) associated with Laplacian on a Lie group, Coll. Math. 31 (1974), 259-287.

[18] A. Hulanicki, J.W. Jenkins, Nilpotent Lie groups and summability of eigenfunction expansions of Schrödinger operators, Studia Math. 80 (1984) 235-244.

[19] G. Mauceri, S. Meda, Vector-Valued Multipliers on Stratified Groups, Revista Math. Iberoamericana (6) 141-154.

[20] D. Müller, E. M. Stein, On spectral multipliers for Heisenberg and related groups, J. de Math. Pure et Appl. 73(1994) 413-440.

[21] T. Pytlik, Symbolic calculus on weighted group algebras, Studia. Math. 73 (1982), 169–176.

[22] A. Seeger, Estimates near L^{1} for Fourier multipliers and maximal functions, Arch. Math. 53 (1989), 188–193.

[23] C. Sogge, On the convergence of Riesz means on compact manifolds, Ann. of Math. 126 (1987), 439–447.

[24] M. Taylor, Pseudodifferential operators, Princeton University Press, Princeton 1981.