Spectral multipliers on metabelian groups

by Waldemar Hebisch

Introduction

Let G be a Lie group, X_{j} right invariant vector fields on G, which generate (as a Lie algebra) the Lie algebra of G,

L = − ∑ X_{j}^{2}.

Then L is called sublaplacian, and it well-known that L is positive definite and essentially selfadjoint on C_{c}^{∞}(G). By the spectral theorem, for any bounded Borel measurable function F : [0, ∞) ↦ ℂ the operator F (L)f = ∫_{0}^{∞}F (λ)dE(λ)f is bounded on L^{2}(G). We are interested in the behavior of F (L) on L^{p}.

This question has long history. Classical results for polynomial growth case are [15], [16], [5], [18], [1], [7], [22] for exponential growth [8], [23], [2], [3]. Newer results show that connection with growth is more complicated [11], [19], [12], [10], [9], [4], [6], [13], [20], [14], [21], [17].

In this paper we consider L^{1}(G) boundedness of F (L) for (some) metabelian G and distinguished L on G. Of the main interest is that the group is of exponential growth, and possibly higher rank. Previously only results about higher rank groups where about Iwasawa type groups. Also, our groups may be unimodular, so it is the second positive result about unimodular groups, and the first giving a family of examples.

Results

Let G = ℝ^{k} ⋉ ℝ^{m}, adjoint action is semisimple, L = L_{0} + L_{1}, L_{0} lives on ℝ^{k}, L_{1} lives on ℝ^{m} and is a sum of (squares of) eigenvectors for adjoint action. More precisely, assume that λ_{j}, j = 1, …, m are linear forms on ℝ^{k}, e_{j}, j = 1, …, m is the canonical basis of ℝ^{m}, linear operator A(x) : ℝ^{m} ↦ ℝ^{m} is given by the formula A(x)e_{j} = λ_{j}(x)e_{j} and

(x_{1}, y_{1})(x_{2}, y_{2}) = (x_{1} + x_{2}, exp(A(−x_{2}))y_{1} + y_{2}).

The right-invariant vector fields are:

X_{j} = ∂_{x_{j}}

and

Y_{j} = exp(−λ_{j}(x))∂_{y_{j}}.

We assume that

L = − ∑ X_{j}^{2} − ∑ Y_{j}^{2} = L_{0} + L_{1}.

We can transform general L_{0} to our form, but for L_{1} the assumption is somewhat restrictive. In this paper we identify convolution operators with functions:

exp(−tL)f = exp(−tL) ∗ f.

(1.1). Theorem. If G and L is as above, then there exist C such that

∥ exp(−(1 + is)L)∥_{L^{1}} ≤ C(1 + |s|^{3m+n}).

(1.2). Theorem. For every compactly supported F ∈ C^{3m+n+1} the operator F (L) is bounded on L^{1}(G).

Theorem (1.2) is a straightforward consequence of (1.1) .

Before the proof we need a lemma about “symbols”. We consider it as well-known, but the form given below is adjusted to our needs.

(1.3). Lemma. There is C such that if E is a normed space, f : ℝ ↦ E, b ≥ 1,

sup |f̂| ≤ a,
sup |ω∂_{ω}f̂(ω)| ≤ ab,
sup |ω^{2}∂_{ω}^{2}f̂(ω)| ≤ ab^{2},

then

|f(x)| ≤ \frac{C}{|x|}.

P. Let ϕ ∈ C^{∞}(ℝ) be such that ϕ(x) = 1 for |x| ≤ 1 and ϕ(x) = 0 for |x| ≥ 2. Fix x_{0} ≠ 0 and let r = \frac{b}{|x_{0}|}. Put f̂_{1}(ω) = ϕ(ω/r)f̂(ω) and f̂_{2}(ω) = (1 − ϕ(ω/r))f̂(ω). We have

|f_{1}(x)| ≤ ∫ |f̂_{1}|dω ≤ ∫_{−2r}^{2r} adω = 4ar,

and

|x^{2}f_{2}(x)| ≤ ∫ |∂_{ω}^{2}f̂_{2}(ω)|dω.

By Leibnitz formula

∂_{ω}^{2}f̂_{2}(ω) = (1 − ϕ(ω/r))∂_{ω}^{2}f̂(ω) − 2r^{−1}ϕ′(ω/r)∂_{ω}f̂(ω) + r^{−2}ϕ′′(ω/r)f̂(ω)

so

∫ |∂_{ω}^{2}f̂_{2}(ω)|dω ≤ ∫_{|ω|>r} \frac{ab^{2}}{ω^{2}} dω + ∫_{2r>|ω|>r} 2Cr^{−1}abω^{−1}dω + ∫_{2r>|ω|>r} Cr^{−2}adω

≤ ab^{2}r^{−1} + 4Cabr^{−1} + 2Car^{−1} ≤ C′ab^{2}r^{−1}.

Now

|f(x_{0}) ≤ |f_{1}(x_{0})| + |f_{2}(x_{0})| ≤ 4ar + C′ab^{2}r^{−1}|x_{0}|^{−2} = (4 + C′)ab|x_{0}|^{−1} = C′′ab|x_{0}|^{−1}.

△

P. of (1.1) . We decompose the regular representation of G using Fourier transform in y variable. In coordinates

L = −Δ_{x} − ∑ exp(2λ_{j}(x))∂_{y_{j}}^{2}

where Δ_{x} = ∑ ∂_{x_{j}}^{2}.

If we denote by H_{z} the Fourier transform (in y variable) of L at z, then

H_{z} = −Δ_{x} + ∑ z_{j}^{2} exp(2λ_{j}(x)).

ℜH_{z} ≥ 0, provided that ℜz_{j} > ℑz_{j}, j = 1, …, m, so z ↦ exp(−tH_{z}) is bounded holomorphic in the area given by the inequalities.

Considering (t + is)H_{z} we see that exp(−(t + is)H_{z}) is bounded and holomorphic as long as ℜ(t + is)z_{j}^{2} ≥ 0, j = 1, …, m. Moreover, we can estimate the integral kernels

∥ exp(−(2t + is)H_{z})δ_{0}∥_{L^{2}} ≤ ∥ exp(−(t + is)H_{z})∥∥ exp(−tH_{z})δ_{0}∥_{L^{2}}.

By the Feynmann-Kac formula

∥ exp(−tH_{z})δ_{0}∥_{L^{2}} ≤ ∥q_{t}∥_{L^{2}} = ct^{−n/4}

where q_{t} is ordinary euclidean heat kernel.

Consequently, by the Cauchy integral formula (for real z)

∥∂_{z}^{α} exp(−(t + is)H_{z})δ_{0}∥_{L^{2}} ≤ C_{α}|z_{1}|^{−α_{1}} · . . . · |z_{m}|^{−α_{m}}(1 + \frac{|s|}{t})^{|α|}t^{−n/4}

Applying (1.3) m times we get

∥ exp(−(t + is)L)(·, y)∥_{L^{2}} ≤ C′′(|y_{1}| · . . . · |y_{m}|)^{−1}(1 + \frac{|s|}{t})^{m}t^{−n/4}.

In [14] (as first step in proof of Theorem (1.1)) we proved that

∫ | exp(−(1 + is)L)(g)|e^{d(g,0)}dg ≤ C exp(Cs^{2})

where d(x, y) is the optimal control distance associated to L. One easily checks that

{g : d(g, 0) < r} ⊂ {(x, y) : |x| < r, |y| < c_{d} exp(c_{d}r)}.

To estimate L^{1} norm we put r = Cs^{2}, c = c_{d}C, A_{j} = {(x, y) : |x| < Cs^{2}, |y_{j}| < exp(−mcs^{2}), |y_{l}| < exp(cs^{2}), l ≠ j}. Note |A_{j}| ≤ Cs^{2n}. We have

∥ exp(−(1 + is)L)∥_{L^{1}} ≤ ∫_{d(g,0)>r} | exp(−(1 + is)L)(g)|dg

+ ∫_{|x|<cs^{2},exp(−mcs^{2})≤|y_{j}|≤exp(cs^{2})} | exp(−(1 + is)L)((x, y))|dxdy

+ ∑_{j} ∫_{A_{j}} | exp(−(1 + is)L)|(g)dg

= I_{∞} + I_{0} + ∑ I_{j}.

For I_{∞} we use exponential estimate

∫_{d(g,0)>r} | exp(−(1 + is)L)(g)|dg ≤ e^{−r} ∫ | exp(−(1 + is)L)(g)| exp(d(g, 0))dg

≤ exp(−Cs^{2})C exp(cs^{2}) = C.

Next

I_{j} ≤ |A_{j}|^{1/2}∥ exp(−(1 + is)L)∥_{L^{2}} ≤ C|s|^{n}.

Finally

I_{0} = ∫_{exp(−mcs^{2})≤|y_{j}|≤exp(cs^{2})} ∫_{|x|<Cs^{2}} | exp(−(1 + is)L)(x, y)|dxdy

≤ ∫_{exp(−mcs^{2})≤|y_{j}|≤exp(cs^{2})} |{x : |x| < cs^{2}}|^{1/2}∥ exp(−(1 + is)L)(·, y)∥_{L^{2}}dy

≤ ∫_{exp(−mcs^{2})≤|y_{j}|≤exp(cs^{2})} cs^{n}C′′(|y_{1}| · . . . · |y_{m}|)^{−1}(1 + |s|)^{m}dy

≤ C|s|^{n}(1 + |s|)^{m} (2 ∫_{exp(−mcs^{2})}^{exp(cs^{2})} |y_{1}|^{−1}dy_{1})^{m}

≤ C|s|^{n}(1 + |s|)^{m}((m + 1)cs^{2})^{m} ≈ C′(1 + |s|^{n+3m}).

△ Final remarks Our goal was to present the idea, so we used simple arguments even though we got

weaker end result. If the estimates are done in more involved way one may replace n + 3m in (1.1) by smaller number (we checked that (n+3m)/2 is enough), however we expect that in (1.2) it is enough to have more than n/2+m derivatives in L^{2}, and getting this requires new ideas. Also, constants in (1.2) grow exponentially with diameter of support of F. We may get polynomial growth, but we would like to have uniform bound on ∥F (tL)∥_{L^{1}}.

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