Analysis of Laplacians on solvable Lie groups.

by Waldemar Hebisch

Institute of Mathematics, Wrocław University, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.

e-mail: hebisch@math.uni.wroc.pl
www: http://www.math.uni.wroc.pl/~hebisch

Notes for Instructional Conference in Edinburgh

General setup Let G be a connected 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}.

If X_{j} linearly span the Lie algebra of G, then L is (up to first order term) the Laplace-Beltrami operator (written in a funny way) for some Riemmanian metric on G. In general L is a hypoelliptic second order operator. L generates convolution semigroup on all L^{p}, p ≥ 1 (L^{p} is with respect to left-invariant Haar measure)

e^{−tL}f = p_{t} ∗ f

p_{t} are called heat kernel, p_{t} > 0, p_{t} ∈ C^{∞}(G), p_{t+s} = p_{t} ∗ p_{s}. They solve the heat equation, namely if f is in L^{2}, u(x, t) = p_{t} ∗ f, then

−Lu = ∂_{t}u

lim_{t→0} u(t, ·) = f in L^{2}.

u is the unique solution which satisfies the initial condition and has uniformly (in t) bounded L^{2} norms on G × {t}. The closure A of linear span of p_{t}, for t > 0 is a (commutative) subalgebra of L^{1} called the subalgebra generated by L. A is a ∗-Banach algebra. Example If L is minus Laplacian on ℝ^{n}, then A is just set of radial L^{1} functions.

L is positive definite and self-adjoint on L^{2}(G) with respect to left Haar measure. By the spectral theorem

F (L) = ∫_{0}^{∞}F (λ)dE(λ)

is well defined operator on L^{2}(G). Spectral decomposition of L and Gelfand transform on A are closely related. For each λ ∈ sp(L) there is a smooth positive definite function ϕ_{λ} such that mapping u ↦ ⟨u, ϕ_{λ}⟩ is a multiplicative functional on A and Lϕ_{λ} = λϕ_{λ}. If G is non-compact amenable group then sp(L) = ℝ_{+}. If u ∈ A and the Gelfand transform of u in ϕ_{λ} equals F (λ), then u ∗ f = F (L)f. In general F (L)f = u ∗ f where u is a distribution — in the sequel we will identify F (L) with the distribution u. We have Plancherel formula: there is a measure μ on ℝ_{+} such that if (the function identified with) F (L) ∈ L^{2} then

∥F (L)∥_{L^{2}}^{2} = ∫ |F (λ)|^{2}dμ(λ).

There is also inversion formula

F (L)(x) = ∫ ϕ_{λ}(x)F (λ)dμ(λ).

μ is locally finite and polynomialy growing. Remark On ℝ^{n} Gelfand transform is basicaly the Fourier transform

{ \atop F (L)}(ω) = F (|ω|^{2}).

We begun from the heat eqation, but we may also consider the wave equation on G × ℝ

Lu = −∂_{t}^{2}u.

If u(0, ·) = f and ∂_{t}u(0, ·) = h, then

u(t, ·) = cos(\sqrt{L})f + \frac{sin(\sqrt{L})}{\sqrt{L}}h.

Two operators on non-unimodular groups On non-unimodular groups there is another operator Δ built from X_{j} and closly related to L. For Δ the natural space is L^{2} with respect to right-invariant Haar measure d_{r}g. We may define Δ by the formula

⟨Δf, f⟩_{r} = ∑⟨X_{j}f, X_{j}f⟩_{r}

where r means that the scalar product is with respect to d_{r}g. d_{r}g is related to the left Haar measure dg via the modular function m:

d_{r}g = mdg

Then we check that

Δ = − ∑(X_{j} + \frac{X_{j}m}{m})X_{j} = L − ∑ \frac{X_{j}m}{m}X_{j}.

Mapping f ↦ Uf = m^{1/2}f is an isometry of L^{2}(d_{r}g) with L^{2}(dg). We write

m^{−1/2}Lm^{1/2} = − ∑(m^{−1/2}X_{j}m^{1/2})^{2} = − ∑(X_{j} + \frac{X_{j}m}{2m})^{2}

= L − ∑ \frac{X_{j}m}{m}X_{j} − \frac{1}{4} ∑ (\frac{X_{j}m}{m})^{2} = Δ − \frac{1}{4} ∑ (\frac{X_{j}m}{m})^{2} .

Hence, U intertwines the spectral resolutions of L and Δ − c where c = \frac{1}{4} ∑ (\frac{X_{j}m}{m})^{2}. If e^{−tΔ}f = q_{t} ∗ f, then

p_{t} = m^{1/2}e^{−tc}q_{t}.

Finite propagaton speed for wave equation There is a natural metric d (optimal control metric) on G associated to L. For this metric we have

supp cos(t\sqrt{L}) ⊂ B(r).

We have also finite propagation speed for Δ. This is related to L by the formula

cos(t\sqrt{L}) = m^{1/2}(cos(t\sqrt{Δ − c})).

(1.1). Theorem. If F is holomorphic and symmetric, such that

|F (z)| ≤ Ce^{rℑz},
∫ |F (\sqrt{(λ)}|^{2}dμ < ∞,

then u = F (\sqrt{L}) ∈ L^{2}(G) and supp u ⊂ B(r). Conversly, if G admits nontrivial homomorphizm into ℝ, u ∈ \overline{L^{2} ∩ A}, supp u ⊂ B(r), then u = F (\sqrt{L}) with F as above. P. By euclidean Paley-Winer theorem supp F̂ ⊂ [−r, r]. Now

F (\sqrt{L}) = ∫ F̂(t) cos(t\sqrt{L})dt

so the first claim follows from finite propagation speed. For the converse note, that u = F (\sqrt{L}) with some F ∈ L^{2}(μ). Now, we take image under homomorphizm and again apply euclidean Paley-Winer.

Questions The basic question is: Q: How does F (L) behave on L^{p}, p ≠ 2. Remark. We are interested in real-variable F so L must have real spectrum. The spectrum of Δ on L^{p} is complex for non-unimodular G and p ≠ 2. Below is an atempt to present state of art, giving more specific questions, and shortly summarizing known answers. Q1: Is A symmetric (if x = x^{∗}, then sp(x) is real). yes if L^{1}(G) is symmetric (for exponential G, D. Poguntke [24] gave characterization), M. Christ and D. Müller [7] show G and L with non-symmetric A, J. Ludwig and D. Müller [19] give a large class of counter-examples Q2: Is it true that:

∃_{k}F ∈ C_{c}^{k}(ℝ_{+}) ⇒ F (L) ∈ L^{1}(G)

yes if dim(G/[G, G]) = 1, real parts of roots are positive yes if G is Iwasawa type, L distinguished [H], [CGHM] (as pointed out by D. Poguntke, L^{1}(G) is usually non-symmetric) yes if 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 eigenvectors for adjoint action (unpublished). no on Heisenberg group extended by hyperbolic dilations [7]. Q3: Is it true that:

∃_{a,b,C,k} supp F ⊂ [a, b] ⇒ sup ∥F (tL)∥_{L^{1}} ≤ C∥F ∥_{C^{k}}

yes if G is Iwasawa type, L distinguished (implicit in [11]) yes for groups considerd in this paper Q4:Is it true that:

∃_{p≠2,C,k,ϕ}ϕ ∈ C_{c}^{∞}(ℝ_{+}), ∥F (L)∥_{L^{p}→L^{p}} ≤ C sup ∥ϕF (t·)∥_{C^{k}}

completely open — there is no Calderon-Zygmund theory on exponential growth groups, newertheless, we expect some positive results. Q5: Find smallest k if the answer for one of the previous questions is positive the critical index at infinity which we get in this paper is the best possible.

Assume that G = A ⋉ N, L = X_{1}^{2} + L_{1}, X_{1} generates A, L_{1} lives on N and [X_{1}, L_{1}] = 2L_{1}. We consider N as a homogeneous group with dilations D_{t}(x) = exp(ctX_{1})x exp(−ctX_{1}), c = log(2)/2. Note, that

D_{k}L = 2^{k}L.

N may be considerd as a stratified nilpotent Lie group, namely, its Lie algebra (which may be identified with N via exponential map) is a stratified nilpotent Lie algebra. That means there is q (we say that N is of step q) such that

N = ⨁_{j=1}^{q} V_{j} ,

V_{1} generates N and [V_{j}, V_{i}] ⊂ V_{i+j} for every 1 ≤ j, i ≤ q. Usually, one assumes that D_{k} is a diagonal operator, and V_{j} are eigenspaces, but our assumption is equivalent to saying that D_{1} = \sqrt{2}U, where U is isometry with respect to the scalar product on V_{1} associated to L. The homogeneous dimension Q of N is defined by the formula

|D_{2k}(A)| = 2^{kQ}|A|

for any measurable A ⊂ N (of course Q = tr_{N}(ad(X_{1})) = ∑ j dim(V_{j})). G as a set is a product of A ≈ ℝ and N with multiplication given by the formula

(s_{1}, n_{1})(s_{2}, n_{2}) = (s_{1} + s_{2}, D_{−s_{2}/c}n_{1} · n_{2}).

For further reference note that the Lebesque measure is left invariant and the modular function m is given by the formula m(s, n) = e^{Qs}.

(1.2). Theorem. Let F_{t}(λ) = F (tλ). If s_{0} > \frac{Q+1}{2}, s_{1} > \frac{3}{2},

∫_{t>1} ∥ϕF_{t}∥_{H(s_{1})} \frac{dt}{t} < ∞

and

∫_{t≤1} ∥ϕF_{t}∥_{H(s_{0})} \frac{dt}{t} < ∞

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

Remark. From (1.2) and trivial L^{2} estimate one can easily interpolate an L^{p} theorem, which however does not seem to be sharp.

(1.3). Lemma. Let q, Q ∈ ℝ_{+}, q ≤ Q, supp f ⊂ [2^{k−1}, 2^{k+1}], f ∈ H(s), s ≥ 0, then there exist symmetric functions f_{l}, such that

f = ∑ f_{l} on ℝ_{+},
supp f̂_{l} ⊂ [−2^{l−k}, 2^{l−k}],

for k ≥ 1,

∫_{0}^{∞}|f_{l}(x)|^{2}(x^{2q} + x^{2Q})dx ≤ C2^{−2sl}2^{(2q+1)k}∥δ_{2^{k}}f∥_{H(s)}^{2}

and for k ≤ 1,

∫_{0}^{∞}|f_{l}(x)|^{2}(x^{2q} + x^{2Q})dx ≤ C2^{−2sl}2^{(2Q+1)k}∥δ_{2^{k}}f∥_{H(s)}^{2}

P. We may assume that k = 0 (otherwise we replace f by δ_{k}f). Next, we extend f symmetricaly to negative halfline. We choose ϕ ∈ C_{c}^{∞}(ℝ) such that ϕ(x) = 1 for |x| < \frac{1}{2} and ϕ(x) = 0 for |x| ≥ 1. We prescribe the Fourier transform of f_{l}:

f̂_{0} = ϕf̂,

and for l > 0

f̂_{l} = (ϕ(2^{−l}x) − ϕ(2^{−l+1}x)) f̂.

By definition we have correct support of f̂_{l} and

∥f_{l}∥_{L^{2}} ≤ C2^{−sl}∥f∥_{H(s)}.

We write

f_{l} = (ψ_{l} − ψ_{l−1}) ∗ f

where ψ̂ = ϕ and ψ_{l}(x) = 2^{l}ψ(2^{l}x). Since supp f ⊂ [−2, 2] and

|ψ_{l}(x) ≤ C2^{l}(1 + 2^{l}|x|)^{−Q−s−1}

so

∫_{|x|>4} |f_{l}|^{2}(1 + |x|)^{2Q} ≤ (∫ |f|)^{2} C ∫_{|x|>4} 2^{2l}(2^{l}(|x| − 2)^{−2Q−2s−2}(1 + |x|)^{2Q}

≤ C∥f∥_{H(s)}^{2}I

Now (|x| − 2) ≥ 1 so

(2^{l}(|x| − 2))^{−2s} ≤ 2^{−2sl}.

Moreover

(2^{l}(|x| − 2))^{−2Q}(1 + |x|)^{2Q} ≤ C.

Finally

I ≤ C2^{−2sl} ∫_{|x|>4} 2^{2l}2^{−2l}(|x| − 1)^{−2} ≤ C2^{−2sl}

Homogeneous estimates We fix a homogeneous norm on N, that is a function | · | on N such that

|x| ≥ 0,  |x| = 0 ⟺ x = 0  |D_{2k}(x)| = 2^{k}|x|.

We put w(s, n) = |n|^{Q}. We will consider w both as function on N and as a function on G (independent on the coordinate from A). In this section we consider functions on N, so we will simply write L instead of L_{1}. On N since L is homogeneous μ is homogeneous too:

∥F (\sqrt{L})∥_{L^{2}}^{2} = c ∫_{0}^{∞}|F (x)|^{2}x^{Q−1}dx.

(1.4). Theorem. If supp f ⊂ [1/2, 2], ε > 0, then

∥f(\sqrt{L})∥_{L^{1}} ≤ C∥f(\sqrt{L})∥_{L^{2}(1+|x|^{Q+ε})} ≤ C′∥f∥_{H( \frac{Q}{2} +ε)}.

P. Apply (1.3) and Plancherel on N.

(1.5). Lemma. On N

∥ ∑ D_{k}f_{k}∥_{L^{2}(w)}^{2} ≤ C_{ε} ∑ ∥f_{k}∥_{L^{2}(1+|x|^{Q+ε})}^{2}

P. D_{k}f_{k} are almost orthogonal in L^{2}(w). Let J = x∂_{x} be generator of dilations on ℝ^{1}.

(1.6). Lemma. If s > \frac{Q}{2}, n > \frac{Q}{2}, f(t) → 0 when t → ∞, then

∥f(L)∥_{L^{2}(w)}^{2} ≤ C ∫ ∥ϕf(t·)∥_{H(s)}^{2} \frac{dt}{t},

∥f(L)∥_{L^{2}(w)}^{2} ≤ C ∫ (|J^{n}f|^{2}(t) + |Jf|^{2}(t)) \frac{dt}{t}.

P. To prove the first inequality, let ϕ be a C^{∞} function on ℝ_{+} such that ∑ ϕ(2^{k}t) = 1 for t > 0. Put f_{k}(t) = f(2^{k}t)ϕ(t). We have

f(L) = ∑ f_{k}(2^{k}L) = ∑ D_{k}f_{k}(L)

and

∑ ∥f_{k}∥_{H(s)}^{2} ≤ C ∫ ∥ϕf(t·)∥_{H(s)}^{2} \frac{dt}{t}

However, one can show that

∥f_{k}(L)∥_{L^{2}(1+|x|^{Q+ε})}^{2} ≤ C∥f_{k}∥_{H(s)}^{2}

so the first inequality follows from (1.5) . Put c_{k} = f(2^{k+1}) − f(2^{k}).

|c_{k}|^{2} = | ∫_{2^{k}}^{2^{k+1}}Jf \frac{dt}{t}|^{2} ≤ ∫_{2^{k}}^{2^{k+1}}|Jf|^{2} \frac{dt}{t} ∫_{2^{k}}^{2^{k+1}} \frac{dt}{t} ≤ log(2) ∫_{2^{k}}^{2^{k+1}}|Jf|^{2} \frac{dt}{t}

so

∑ |c_{k}|^{2} ≤ log(2) ∫ |Jf|^{2} \frac{dt}{t}.

Let ψ be a C^{∞} function on ℝ_{+} such that ψ(t) = 0 for t ≥ 2 and ψ(t) = 1 for t ≤ 1. Put g(t) = − ∑ c_{k}ψ(2^{−k}t), h = f − g. To see that series defining g is convergent note that for given t there is k_{0} such that terms with index k < k_{0} are equal to 0 and terms with index k > k_{0} are equal to c_{k}. Also ∑_{k=l}^{m} c_{k} = f(2^{m+1}) − f(2^{l}). Since f(t) goes to 0 when t goes to ∞, the series of c_{k} is convergent. This argument also shows that for integral l we have g(2^{l}) = f(2^{l}). Now

g(L) = − ∑ c_{k}D_{−k}ψ(L)

and (by (1.5) )

∥g(L)∥_{L^{2}(w)}^{2} ≤ C ∑ |c_{k}|^{2} ≤ C′ ∫ |Jf|^{2} \frac{dt}{t}.

Next, h(2^{k}) = 0 so

∫_{2^{k}}^{2^{k+1}}|h|^{2} \frac{dt}{t} ≤ ∫_{2^{k}}^{2^{k+1}}(∫_{2^{k}}^{t} |Jh|(s)\frac{ds}{s})^{2} \frac{dt}{t}

≤ log(2) ∫_{2^{k}}^{2^{k+1}}∫_{2^{k}}^{t} |Jh|^{2}(s)\frac{ds}{s} \frac{dt}{t}

≤ log(2)^{2} ∫_{2^{k}}^{2^{k+1}}|Jh|^{2}(t)\frac{dt}{t}

so

∫ |h|^{2} \frac{dt}{t} ≤ log(2)^{2} ∫ |Jh|^{2}(t)\frac{dt}{t}.

Now, we get the second inequality applying the first inequality to h.

(1.7). Lemma. If N = ℝ^{2n}, L = ∑_{j=1}^{2n} ∂_{j}^{2}, then

∥f(L)∥_{L^{2}(w)}^{2} ≥ c ∫_{0}^{∞}(|J^{n}f|^{2} + |Jf|^{2}) \frac{dt}{t}

P. One easily checks that left side has form

∫ | ∑_{k=1}^{n} c_{k}J^{k}f|^{2} \frac{dt}{t}.

For n = 1 this gives the claim. Let L_{n} be laplacian on ℝ^{2n}. Strightforward calculation shows that

∥f(L_{n})∥_{L^{2}(w)}^{2} ≤ C_{n}∥f(L_{n+1})∥_{L^{2}(w)}^{2}.

Now we proceed by induction on n: we get the estimate for lower order terms from the inductive assumption. AN groups The distance naturally associated to L (optimal control metric) for our groups is given by the formula

d(s, n) = arccosh(\frac{e^{−2s} + 1 + |n|^{2}}{e^{−s}})

provided that |n| is the optimal control distance between n and e associated to L_{1} on N. Put B_{r} = {x ∈ G : d(x, e) < r}.

(1.8). Lemma.

∫_{B_{r}} (1 + w)^{−1} ≤ r^{2}

For Iwasawa AN groups we know the Plancherel measure of L — there is a μ such that ∥f(L)∥^{2} = ∫ |f(λ)|^{2} dμ(λ)

where dμ = h(λ)dx, h is bounded by a polynomial and h(λ) ≈ λ^{\frac{1}{2}} for small λ (cf. for example [11]). What we need is the following estimate

∥f(\sqrt{L})∥^{2} ≤ C ∫ |f(λ)|^{2}(λ^{2} + λ^{Q})dλ.

Main property of Iwasawa AN group is that Δ is invariant under a large group – in the rank 1 case isometries are transitive on spheres. This implies that functions of Δ are radial, and that for L we have

F (L)(x) = m^{1/2}ν(x)

where ν is radial. (1.9). Lemma.

∫_{B_{r}} |f(L)|^{2}w ≤ Cr ∫_{B_{r}} |f(L)|^{2}

P. ψ = f(L) has the following form

ψ(s, n) = e^{\frac{−Qs}{2}} ν(d((s, n), e))

where ν is function on ℝ_{+}. On most of the annulus r −1 < d((s, n), e) < r we approximatly have

|n| ≈ e^{\frac{r−s}{2}}

(more precisely |n| is a function of r and s, if d((s, n), e) < r then |n| < e^{\frac{r−s}{2}} , and if |s| < r − 1, d((s, n), e) = r, then |n| > ce^{\frac{r−s}{2}} ), so

∫_{r−1<d((s,n),e)<r} ψ^{2} ≈ ∫_{r−1}^{r} ν^{2} ∫_{−r}^{r} e^{Qs}e^{\frac{Q(r−s)}{2}} ≈ e^{Qr} ∫_{r−1}^{r} ν^{2}

and

∫_{r−1<d((s,n),e)<r} ψ^{2}w ≈ ∫_{r−1}^{r} ν^{2} ∫_{−r}^{r} e^{Qs}e^{Q(r−s)} ≈ re^{Qr} ∫_{r−1}^{r} ν^{2}

Let A_{1} be the subalgebra of L^{1}(N) generated by L_{1}.

(1.10). Lemma. If f(λ) = ∑ c_{k}e^{−t_{k}λ}, t_{k} > 0, then f(L) ∈ L^{1}(ds, A_{1}), moreover the Gelfand transform of f(L) along N is independent of N

P. First consider N = ℝ. Mapping (s, n) ↦ (s, −n) is an automorphism of G preserving L, so p_{t} is symmetric in n. Put

h_{t}(s, λ^{2}) = ∫ p_{t}(s, n)e^{iλn}dn.

h_{t}(s, λ) is defined for λ ≥ 0. Since p_{t} is holomorphic semigroup on L^{1}

∂_{t}h_{t}(s, λ) = ∂_{s}^{2}h_{t}(s, λ) − e^{−2s}λh_{t}(s, λ),

∫ sup_{λ} |h_{t}(s, λ)| ds = 1

and

lim_{t→0_{+}} h_{t}(s, λ) = δ(s).

We return now to general N. It is easy to see that putting

T (t)(s, n) = h(s, L_{1})(n)

we get continuous family of contractions on L^{2}(G). Moreover

∂_{t}T (t) = (∂_{s}^{2} − e^{−2s}L_{1})T (t) = LT (t),

lim_{t→0_{+}} T (t) = I

hence T (t) = e^{−tL}. This gives the conclusion for p_{t}, that is for f(λ) = e^{−tλ}, and than we get the claim by linearity.

(1.11). Lemma. If supp f̂ ⊂ [−r, r], then for r > 1, and even Q,

∥f(\sqrt{L})∥_{L^{1}} ≤ Cr^{3/2} (∫_{0}^{∞}|f(x)|^{2}(x^{2} + x^{Q+2})dx)^{1/2}

and for r ≤ 1,

∥f(\sqrt{L})∥_{L^{1}} ≤ Cr^{(Q+1)/2} (∫_{0}^{∞}|f(x)|^{2}(x^{2} + x^{Q})dx)^{1/2}

P. If r ≤ 1, then

∥f(\sqrt{L})∥_{L^{1}} ≤ ( ∫_{|x|<r} 1)^{1/2} ∥f(\sqrt{L})∥_{L^{2}}

≤ Cr^{(Q+1)/2} (∫_{0}^{∞}|f(x)|^{2}(x^{2} + x^{Q})dx)^{1/2} .

If r > 1, then Q is even. Next, let L̃ be the laplacian on Iwasawa type AN with N = ℝ^{Q+2}. We have

∥f(\sqrt{L})∥_{L^{1}} ≤ ( ∫_{|x|<r} \frac{1}{1 + w} )^{1/2} ∥f(\sqrt{L})∥_{L^{2}(1+w)}

Now by (1.6) , (1.7) , (1.10) and (1.9)

∥f(\sqrt{L})∥_{L^{2}(1+w)}^{2} ≤ C_{1}∥f(\sqrt{L̃})∥_{L^{2}(1+w)}^{2}

≤ C_{2}r∥f(\sqrt{L̃})∥_{L^{2}}^{2}

so

∥f(\sqrt{L})∥_{L^{1}} ≤ Cr^{3/2} (∫_{0}^{∞}|f(x)|^{2}(x^{2} + x^{Q+2})dx)^{1/2}

Proof of (1.2) : We split F. Let ψ be C_{c}^{∞} function such that supp ψ ⊂ [1/2, 2] and for all x > 0 ∑ ψ(2^{k}x) = 1. We write F (x) = ∑ F_{k}(x), where F_{k}(x) = m(x)ψ(2^{−k}x). It is enough to prove that for k ≤ 0 and s > 3/2 we have

∥F_{k}(\sqrt{L})∥_{L^{1}} ≤ C∥δ_{2^{k}}F_{k}∥_{H(s)}

and that for k ≥ 0 and s > (Q + 1)/2 we have

∥F_{k}(\sqrt{L})∥_{L^{1}} ≤ C∥δ_{2^{k}}F_{k}∥_{H(s)}.

Fix k ≤ 0. We split f = F_{k} using (1.3) (with Q replaced by even number greater then Q + 2). By (1.11)

∥f_{l}(\sqrt{L})∥_{L^{1}} ≤ C(2^{l−k})^{3/2}2^{−sl}2^{3k/2}∥δ_{2^{k}}F_{k}∥_{H(s)}
= C2^{( \frac{3}{2} −s)l}∥δ_{2^{k}}F_{k}∥_{H(s)}

so

∥f(\sqrt{L})∥_{L^{1}} ≤ ∑ ∥f_{l}(\sqrt{L})∥_{L^{1}} ≤ C′∥δ_{2^{k}}F_{k}∥_{H(s)}.

Case k ≥ 0 is similar.

References

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

[2] J. Ph. Anker, L^{p} Fourier multipliers on Riemannian symmetric spaces of the non-compact type, Ann. of Math. 132 (1990), 597-628.

[3] J. Ph. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J. 65 (1992), 257–297.

[4] F. Astengo, Multipliers for a distinguished Laplacean on solvable extensions of H-type groups, Monatsh. Math. 120 (1995), 179–188.

[5] Ph. Bougerol, Examples de théorm̀es locaux sur certains groupes résolubles, Ann. I. H. P. 19 (1983), 369–391.

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

[7] M. Christ, D. Müller, On L^{p} spectral multipliers for a solvable Lie group Geom. Funct. Anal. 6 (1996), 860–876.

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

[9] J. L. Clerc, E. M. Stein, L^{p} -multipliers for non-compact symmetric spaces, Proc. Nat. Acad. Sci. USA (1974), 3911–3912.

[10] M. Cowling, Harmonic analysis on semigroups, Ann. of Math. 117 (1983), 267–283.

[11] M. Cowling, S. Giulini, A. Hulanicki, G. Mauceri, Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth, Studia Math. 111 (1994), 103–121.

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

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

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

[15] W. Hebisch, Boundedness of L^{1} spectral multipliers for an exponential solvable Lie group, Coll. Math. 73 (1997), 155–164.

[16] W. Hebisch, Spectral multipliers on exponential growth solvable Lie groups, Math. Z. 229 (1998), 435–441.

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

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

[19] J. Ludwig, D. Müller, Sub-Laplacians of holomorphic L^{p}-type on rank one ANgroups and related solvable groups, preprint.

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

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

[22] S. Mustapha, Multiplicateurs spectraux sur certains groupes non-unimodulaires, Harmonic Analysis and Number Theory, CMS Conf. Proceedings, Vol 21, 1997.

[23] S. Mustapha, Multiplicateurs de Mikhlin pour une classe particulière de groupes non-unimodulaires, Ann. Inst. Fourier 48 (1998), 957–966.

[24] D. Poguntke, Algebraically irreducible representations of L^{1}-algebras of exponential Lie groups, Duke Math. J. 50 (1983), 1077–1106.

[25] A. Sikora, Multiplicateurs associés aux souslaplaciens sur les groupes homogènes, C.R. Acad. Sci. Paris, Série I, 315 (1992), 417–419.

[26] A. Sikora, Sharp pointwise estimates on heat kernels, Quart. J. Math. 47(1996), 731–382.

[27] E. M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory, Ann. of Math. Stud. 63, Princeton Univ. Press, Princeton 1970.

[28] M. Taylor, L^{p} -Estimate on functions of the Laplace operator, Duke Math. J. 58 (1989), 773-793.