Journal of Lie Theory Volume 5 (1995) 1–14 C ◯1995 Heldermann Verlag

L^{p}–L^{q}–Estimates for functions of the Laplace–Beltrami operator

on noncompact symmetric spaces, II*

Michael Cowling, Saverio Giulini and Stefano Meda

Communicated by J. Faraut

In this paper we continue the study of functional calculus for the Laplace– Beltrami operator on symmetric spaces of the noncompact type begun in [3]; this paper is dedicated to a study of the Poisson semigroup, which we define shortly.

Let G and K be a connected noncompact semisimple Lie group with finite center and a maximal compact subgroup thereof, and consider the symmetric space G/K, also denoted by X. We denote by n the dimension of X, by ℓ its real rank, and by ν the “pseudo–dimension” 2 |Σ_{0}^{+}| + ℓ, where |Σ_{0}^{+}| is the cardinality of the set of the positive indivisible (restricted) roots.

There is a canonical invariant Riemannian metric on X; denote by −L_{0} the associated Laplace–Beltrami operator. By general nonsense, L_{0} is positive and essentially self-adjoint on C_{c}^{∞}(X); let L be the unique self-adjoint extension of L_{0} and {P_{η}} the spectral resolution of the identity for which

Lf = ∫_{R_{0}^{2}}^{∞} η dP_{η}f ∀f ∈ Dom(L),

where R_{0} = ⟨ρ, ρ⟩^{1/2}, ρ being the usual half-sum of the positive roots.

For θ in [0, 1] and σ in (0, 1), the θ–heat and the (σ, θ)–Poisson semigroups (H_{t,θ})_{t>0} and (P_{t,θ}^{σ})_{t>0} are defined thus:

H_{t,θ}f = ∫_{R_{0}^{2}}^{∞} exp(−t(η − θR_{0}^{2})) dP_{η}f  ∀t ∈ (0, ∞) ∀f ∈ L^{2}(X)
P_{t,θ}^{σ}f = ∫_{R_{0}^{2}}^{∞} exp(−t(η − θR_{0}^{2})^{σ}) dP_{η}f  ∀t ∈ (0, ∞) ∀f ∈ L^{2}(X).

The (σ, θ)–Poisson semigroup may be obtained from the θ–heat semigroup by subordination. However, while estimates from above can be proved using this

ISSN 0940–2268 / $2.50 ^{_{C} ◯} Heldermann Verlag fact, estimates from below cannot. If 1 ≤ p, q ≤ ∞ and the operator P_{t,θ}^{σ} satisfies a norm inequality of the form

∗ Work supported by the Australian Research Council and the Italian M.U.R.S.T., fondi 60%.

∥P_{t,θ}^{σ}f∥_{q} ≤ C ∥f∥_{p} ∀f ∈ L^{2}(X) ∩ L^{p}(X),

P_{t,θ}^{σ} is said to be L^{p} –L^{q} -bounded.

In this paper we examine for which p and q the operator P_{t,θ}^{σ} is L^{p} –L^{q} - bounded, and we study the behaviour of the L^{p} –L^{q} -operator norms |||P_{t,θ}^{σ}|||_{p;q} as t tends to 0 and to ∞ for all such p and q. As t tends to ∞, the expression describing the behaviour of |||P_{t,θ}^{σ}|||_{p;q} involves powers of t, in which the indices ν and ℓ play an important rôle. Two features of our study are noteworthy. First, while H_{t,θ} is L^{p} –L^{q} -bounded whenever 1 ≤ p ≤ q ≤ ∞, P_{t,θ}^{σ} is not L^{p} –L^{q} - bounded for many such p and q. Second, when p or q reach the critical index for L^{p} –L^{q} -boundedness, the exponent ℓ + 1 appears; in our previous work, we saw only ℓ. We refer to [3] for an account of related work.

In order to state our main theorem, we introduce a little notation: if 0 ≤ θ ≤ 1,

p_{θ} = 2/(1 + (1 − θ)^{1/2}),
R_{θ} = [(1 − θ) ⟨ρ, ρ⟩]^{1/2},
R_{θ,p} = [(\frac{4}{pp′} − θ) ⟨ρ, ρ⟩]^{1/2};

in the definition of R_{θ,p}, p_{θ} ≤ p ≤ p_{θ}′. Observe that 1 ≤ p_{θ} ≤ 2, that R_{θ} defined here agrees with R_{0} as defined previously when θ = 0, that R_{θ,p} = R_{θ} when p = 2, and that R_{θ,p} = 0 when p = p_{θ} or p = p_{θ}′.

Theorem 1. Suppose that 0 ≤ θ ≤ 1, 0 < σ < 1, and 1 ≤ p, q ≤ ∞. The following conditions hold:

(i) if t > 0, then P_{t,θ}^{σ} is L^{p} – L^{q} -bounded only if p ≤ q, p ≤ p_{θ}′, and q ≥ p_{θ};

(ii) if p_{θ} ≤ p ≤ p_{θ}′, then
|||P_{t,θ}^{σ}|||_{p;p} = exp(−R_{θ,p}^{2σ} t)  ∀t ∈ (0, ∞);
(iii) if p ≤ q, p ≤ p_{θ}′ and q ≥ p_{θ}, then
|||P_{t,θ}^{σ}|||_{p;q} ∼ t^{−n(1/p−1/q)/2σ}  ∀t ∈ (0, 1];
(iv) if p < q = 2 or 2 = p < q, then
|||P_{t,θ}^{σ}|||_{p;q} ∼ t^{−ν/4} exp(−R_{θ}^{2σ}t)  ∀t ∈ [1, ∞);

(v) if p < 2 < q, then
|||P_{t,θ}^{σ}|||_{p;q} ∼ t^{−ν/2} exp(−R_{θ}^{2σ}t)  ∀t ∈ [1, ∞);
(vi) if p < q < 2 and q > p_{θ}, then

|||P_{t,θ}^{σ}|||_{p;q} ∼ t^{−ℓ/2q′} exp(−R_{θ,q}^{2σ} t)  ∀t ∈ [1, ∞);

(vii) if p < q = p_{θ}, then

|||P_{t,θ}^{σ}|||_{p;q} ∼ t^{−(ℓ+1)/2σq′}  ∀t ∈ [1, ∞);

(viii) if 2 < p < q and p < p_{θ}′, then

|||P_{t,θ}^{σ}|||_{p;q} ∼ t^{−ℓ/2p} exp(−R_{θ,p}^{2σ} t)  ∀t ∈ [1, ∞);

(ix) if p_{θ}′ = p < q, then

|||P_{t,θ}^{σ}|||_{p;q} ∼ t^{−(ℓ+1)/2σp}  ∀t ∈ [1, ∞).

Section 1 of this paper is devoted to notation, and a summary of relevant material, such as the the spherical Fourier transformation. In Section 2 we prove our theorem.

1. Notation and Background Material

We use the standard notation of the theory of Lie groups and symmetric spaces, as in, for instance, S. Helgason [6]. Our notation here is also consistent with our paper [3], to which we refer several times.

For any x in G, we denote by A(x) the element of 𝔞 such that x ∈ N exp A(x) K (the Iwasawa decomposition). For any (complex-valued) linear form Λ on 𝔞, the elementary spherical function ϕ_{Λ} is defined by the rule

ϕ_{Λ}(x) = ∫_{K} exp((iΛ + ρ) A(kx)) dk ∀x ∈ G.

The spherical Fourier transform f̃ of an L^{1}(G)–function f is defined by the formula

f̃(Λ) = ∫_{G} f(x) ϕ_{−Λ}(x) dx ∀Λ ∈ 𝔞^{∗}.

Harish–Chandra proved an inversion formula and a Plancherel formula for the spherical Fourier transformation, namely

f(x) = ∫_{𝔞^{∗}} f̃(Λ) ϕ_{Λ}(x) dμ(Λ) ∀x ∈ G

for “nice” K –bi-invariant functions f on G, and

∥f∥_{2} = [∫_{𝔞^{∗}} |f̃(Λ)|^{2} dμ(Λ)]^{1/2} ∀f ∈ L^{2}(K\X),

where dμ(Λ) = c_{G} |c(Λ)|^{−2} dΛ, and c denotes the Harish–Chandra c–function. For the details, see, for instance, Theorem IV.7.5 of Helgason [6]. We often deal with the inversion formula and the Plancherel formula with purely radial integrands. From the formula for the c–function (Theorem IV.6.14 of [6]), it is clear that if F : [0, ∞) → C and f(Λ) = F (|Λ|) for all Λ in 𝔞^{∗}, then

(1) ∫_{𝔞^{∗}} f(Λ) dμ(Λ) = c_{G} ∫_{𝔞^{∗}} F (|Λ|) |c(Λ)|^{−2} dΛ ∼ ∫_{0}^{∞} F (r) (1 + r)^{n−ν} r^{ν−1} dr,

provided the integrals converge. We shall also use a modified version of the Plancherel measure, μ̃, defined by the rule

dμ̃(Λ)/dΛ = ∏_{α∈Σ_{0}^{+}}(1 + |⟨α, Λ⟩|)^{d_{α}}

(see [3] for the notation). Clearly

C ≤ dμ̃(Λ)/dΛ ≤ C′(1 + |Λ|^{n−ℓ}),

where C and C′ are constants. The modified Plancherel measure appears in several results of harmonic analysis on symmetric spaces; see [3].

Let W_{1} be the interior of the convex hull in 𝔞^{∗} of the images of ρ under the Weyl group of (𝔤, 𝔞). For δ in (0, 1), we denote by W_{δ} and T_{δ} the dilate of W_{1} by δ and the tube over the polygon W_{δ}, i.e., T_{δ} = 𝔞^{∗} + iδ W_{1}. If 1 ≤ p < 2, the spherical Fourier transform of an L^{p}(G)-function extends to a bounded holomorphic function in the tube T_{δ(p)}, where δ(p) is defined by the rule

δ(p) = 2/p − 1.

We define the quadratic function Q_{θ} on 𝔞_{C}^{∗} :

Q_{θ}(Λ) = ⟨Λ, Λ⟩ + (1 − θ)⟨ρ, ρ⟩ ∀Λ ∈ 𝔞_{C}^{∗},

and denote by p_{t,θ}^{σ} the K –bi-invariant function on G such that

p_{t,θ}^{σ} (Λ) = exp(−tQ_{θ}(Λ)^{σ}) ∀Λ ∈ T_{δ(p_{θ})}.

Then

P_{t,θ}^{σ}f = f ∗ p_{t,θ}^{σ} ∀t ∈ (0, ∞) ∀f ∈ L^{2}(X).

Note that Q_{θ}^{σ} and p_{t,θ}^{σ} continue analytically to the tube T_{δ(p_{θ})}, but to no larger tube. We denote by h_{t} the kernel associated to the heat operator, i. e., h̃_{t}(Λ) = exp ( − tQ_{0}(Λ)).

Throughout this paper, the following assumptions are made about the parameters:

1 ≤ p, q ≤ ∞,
1/p + 1/p′ = 1,
0 ≤ θ ≤ 1,
0 < σ < 1,
0 < t < ∞.

Recall that p_{θ} = 2/[1+(1−θ)^{1/2}], and note that δ(p_{θ}) = (1−θ)^{1/2}. By C and C′ we denote positive constants which may not be the same at different occurrences; C and C′ may depend on anything quantified, implicitly or explicitly, before the formula in which they appear. The expression

A(t) ∼ B(t) ∀t ∈ D,

where D is some subset of the domains of A and of B, means that there exist constants C and C′ such that

C |A(t)| ≤ |B(t)| ≤ C′ |A(t)| ∀t ∈ D.

If the operator T on L^{2}(X) satisfies a norm inequality of the form

∥T f∥_{q} ≤ C ∥f∥_{p} ∀f ∈ L^{2}(X) ∩ L^{p}(X),

then T extends uniquely to a bounded operator from L^{p}(X) to L^{q}(X) (where the weak-star topology should be used if p = ∞); conversely, if a continuous extension to a bounded operator from L^{p}(X) to L^{q}(X) exists, then such a norm inequality holds. We denote by |||T |||_{p;q} the norm of the linear operator T from L^{p}(X) to L^{q}(X).

2. Estimates for the (σ, θ)–Poisson semigroup

First we prove a couple of technical results on integration, then we estimate the L^{p} –norms of p_{t,θ}^{σ} for various p. Finally we put the ingredients together to prove the main theorem.

Lemma 1. Suppose that 0 ≤ α < β ≤ ∞, ω > 0, and η > 0. Suppose also that ψ is a function on the interval [α, β), which is continuous and strictly positive throughout [α, β), such that, for constants C and k,

|ψ(s)| ≤ C (1 + s)^{k} ∀s ∈ [α, β).

Then

(i) if α = 0,
∫_{α}^{β} exp(−ts^{ω}) (s − α)^{η−1} ψ(s) ds ∼ t^{−η/ω}  ∀t ∈ [1, ∞);
(ii) if α > 0,
∫_{α}^{β} exp(−ts^{ω}) (s − α)^{η−1} ψ(s) ds ∼ t^{−η} exp(−α^{ω}t)  ∀t ∈ [1, ∞).

Proof. We assume initially that β < ∞.

We first prove (i). By changing variables (v = ts^{ω}), we transform the integral to

\frac{t^{−η/ω}}{ω} ∫_{0}^{tβ^{ω}} exp(−v) v^{η/ω−1} ψ((v/t)^{1/ω}) dv;

the integrand is dominated by

exp(−v) v^{η/ω−1} max{|ψ(s)| : s ∈ [α, β)},

which is integrable on R^{+}, and as t tends to ∞ the integral tends to

∫_{0}^{∞} exp(−v) v^{η/ω−1} ψ(0) dv

by the Lebesgue dominated convergence theorem.

We now prove (ii). Let γ denote β^{ω} − α^{ω}. Since ψ on [α, β) is continuous, bounded, and strictly positive, the function ϕ on [0, γ), such that ϕ(0) = ω^{1−η} α^{η(1−ω)} ψ(α) and

ϕ(v) = [\frac{(v + α^{ω})^{1/ω} − α}{v}]^{η−1} ψ((v + α^{ω})^{1/ω}) (v + α^{ω})^{1/ω−1} ∀v ∈ (0, γ),

is too. By changing variables (v = s^{ω} − α^{ω}), we transform the integral to

\frac{exp(−tα^{ω})}{ω} ∫_{0}^{γ} exp(−tv) v^{η−1} ϕ(v) dv.

We have therefore an integral of the form already treated in (i), and are done.

The case where β = ∞ now follows: we write the integral as the sum of an integral from α to α + 1 and one from α + 1 to ∞; the first integral is treated by the result already established, and the second is easily shown to be O(t^{k+η+1} exp(−(α + 1)^{ω}t)). ⊓ ⊔

Lemma 2. Suppose that 0 < τ < ∞ and R > 0. Then

∫_{0}^{1} exp(−t[r^{2} + R^{2}]^{σ}) r^{τ−1} dr ∼ \{ \begin{matrix}1 & ∀t ∈ (0, 1] \\ t^{−τ/2} exp(−R^{2σ}t) & ∀t ∈ [1, ∞) \end{matrix}

and

∫_{1}^{∞} exp(−t[r^{2} + R^{2}]^{σ}) r^{τ−1} dr ∼ \{ \begin{matrix}t^{−τ/2σ} & ∀t ∈ (0, 1] \\ t^{−1} exp(−[1 + R^{2}]^{σ} t) & ∀t ∈ [1, ∞). \end{matrix}

Proof. It is trivial that the first integral behaves as claimed for t in (0, 1]. To study its behaviour for t in [1, ∞), we change variables (s = [r^{2} + R^{2}]^{1/2}), and it becomes

∫_{R}^{√\overline{R^{2}+1}} exp(−ts^{2σ}) (s^{2} − R^{2})^{(τ−2)/2} s ds

= ∫_{R}^{√\overline{R^{2}+1}} exp(−ts^{2σ}) (s − R)^{(τ−2)/2} (s + R)^{(τ−2)/2} s ds.

The required behaviour for t in [1, ∞) is a corollary of Lemma 1 (where ω = 2σ, η = τ/2, α = R, and ψ(s) = (s + R)^{(τ−2)/2} s).

By performing the same change of variables, we transform the second integral to

∫_{√\overline{R^{2}+1}}^{∞} exp(−ts^{2σ}) (s^{2} − R^{2})^{(τ−2)/2} s ds.

The result stated for t in [1, ∞) follows from Lemma 1 (where ω = 2σ, η = 1, α = [R^{2} + 1]^{1/2}, and ψ(s) = (s^{2} − R^{2})^{(τ−2)/2} s). By changing variables again, we transform the last integral to

t^{−τ/2σ} ∫_{\sqrt{R^{2}+1} t^{1/2σ}}^{∞} exp(−v^{2σ}) (v^{2} − (t^{1/2σ}R)^{2})^{(τ−2)/2} v dv

Cowling, Giulini and Meda 7

and the required behaviour for t in (0, 1] is an immediate consequence. ⊓ ⊔ Note that the preceding lemma holds for any positive σ. We now begin the harmonic analysis.

Lemma 3. The following norm estimates for p_{t,θ}^{σ} hold:

(i) if p = 2, then

∥p_{t,θ}^{σ}∥_{p} ∼ \{ \begin{matrix}t^{−n/4σ} & ∀t ∈ (0, 1] \\ t^{−ν/4} exp(−R_{θ}^{2σ}t) & ∀t ∈ [1, ∞); \end{matrix}

(ii) if p = ∞, then

∥p_{t,θ}^{σ}∥_{p} ∼ \{ \begin{matrix}t^{−n/2σ} & ∀t ∈ (0, 1] \\ t^{−ν/2} exp(−R_{θ}^{2σ}t) & ∀t ∈ [1, ∞); \end{matrix}

(iii) if p_{θ} < p < 2, then

∥p_{t,θ}^{σ}∥_{p} ∼ \{ \begin{matrix}t^{−n/2σp′} & ∀t ∈ (0, 1] \\ t^{−ℓ/2p′} exp(−R_{θ,p}^{2σ} t) & ∀t ∈ [1, ∞); \end{matrix}

(iv) if p = p_{θ} < 2, then

∥p_{t,θ}^{σ}∥_{p} ∼ \{ \begin{matrix}t^{−n/2σp′} & ∀t ∈ (0, 1] \\ t^{−(ℓ+1)/2σp′} & ∀t ∈ [1, ∞). \end{matrix}

Proof. To prove (i) we use the Plancherel formula and pass to polar coordinates, using formula (1):

∥p_{t,θ}^{σ}∥_{2} = [∫_{𝔞^{∗}} exp(−2tQ_{θ}(Λ)^{σ}) dμ(Λ)]^{1/2}
∼ [∫_{0}^{∞} exp(−2t[r^{2} + R_{θ}^{2}]^{σ}) (1 + r)^{n−ν} r^{ν−1} dr]^{1/2}
∼ [∫_{0}^{1} exp(−2t[r^{2} + R_{θ}^{2}]^{σ}) r^{ν−1} dr
+ ∫_{1}^{∞} exp(−2t[r^{2} + R_{θ}^{2}]^{σ}) r^{n−1} dr]^{1/2} ∀t ∈ (0, ∞).

Now, from Lemma 2,

∫_{0}^{1} exp(−2t[r^{2} + R_{θ}^{2}]^{σ}) r^{ν−1} dr ∼ \{ \begin{matrix}1 & ∀t ∈ (0, 1] \\ t^{−ν/2} exp(−2R_{θ}^{2σ}t) & ∀t ∈ [1, ∞) \end{matrix}

and

∫_{1}^{∞} exp(−2t[r^{2} + R_{θ}^{2}]^{σ}) r^{n−1} dr ∼ \{ \begin{matrix}t^{−n/2σ} & ∀t ∈ (0, 1] \\ t^{−1} exp(−2[1 + R_{θ}^{2}]^{σ} t) & ∀t ∈ [1, ∞). \end{matrix}

This proves (i).

To prove (ii), we proceed similarly, using the inversion formula. For any x in G,

|p_{t,θ}^{σ}(x)| = |∫_{𝔞^{∗}} exp(−tQ_{θ}(Λ)^{σ}) ϕ_{Λ}(x) dμ(Λ)|
≤ ∫_{𝔞^{∗}} exp(−tQ_{θ}(Λ)^{σ}) |ϕ_{Λ}(x)| dμ(Λ)
≤ ∫_{𝔞^{∗}} exp(−tQ_{θ}(Λ)^{σ}) ϕ_{Λ}(e) dμ(Λ)
= p_{t,θ}^{σ}(e),

so that ∥p_{t,θ}^{σ}∥_{∞} = p_{t,θ}^{σ}(e). Now it is easy to see that

∥p_{t,θ}^{σ}∥_{∞} = ∫_{𝔞^{∗}} exp(−tQ_{θ}(Λ)^{σ}) dμ(Λ) ∼ \{ \begin{matrix}t^{−n/2σ} & ∀t ∈ (0, 1] \\ t^{−ν/2} exp(−R_{θ}^{2σ}t) & ∀t ∈ [1, ∞) \end{matrix}

by a calculation like that of the L^{2}(G)–norm of p_{t,θ}^{σ}.

The hardest parts of this lemma are (iii) and (iv). We prove both cases by obtaining first a lower bound, then an upper bound.

We begin the proof of (iii). Suppose that p_{θ} < p < 2. By Theorem 2.1 of [3], and the fact that dμ̃(Λ)/dΛ ≥ C,

(2) ∥p_{t,θ}^{σ}∥_{p} ≥ C[∫_{𝔞^{∗}} |p_{t,θ}^{σ} (Λ + iδ(p)ρ)|^{p′} dμ̃(Λ)]^{1/p′}

≥ C[∫_{𝔟} exp(−tp′ Re([Q_{θ}(Λ + iδ(p)ρ)]^{σ})) dΛ]^{1/p′} ∀t ∈ [1, ∞),

where 𝔟 denotes the unit ball in 𝔞^{∗}. We denote the right hand side of the last inequality by I(t). Now for all Λ in 𝔞^{∗},

Re([Q_{θ}(Λ + iδ(p)ρ)]^{σ}) ≤ |Q_{θ}(Λ + iδ(p)ρ)|^{σ}
= (⟨Λ, Λ⟩ + R_{θ,p}^{2} + 2iδ(p)⟨ρ, Λ⟩)^{σ/2}
≤ ([⟨Λ, Λ⟩ + R_{θ,p}^{2} ]^{2} + 4δ(p)^{2}R_{0}^{2}⟨Λ, Λ⟩)^{σ/2}
= (⟨Λ, Λ⟩^{2} + γ_{1}⟨Λ, Λ⟩ + γ_{2})^{σ/2},

where γ_{1} = 2R_{θ,p}^{2} + 4δ(p)^{2}R_{0}^{2} and γ_{2} = R_{θ,p}^{4} , so, by using polar co-ordinates, and then changing variables (s = (r^{4} + γ_{1}r^{2} + γ_{2})^{1/4}), we see that

I(t) ≥ C[∫_{0}^{1} exp (−tp′(r^{4} + γ_{1}r^{2} + γ_{2})^{σ/2}) r^{ℓ−1} dr]^{1/p′}

= C[∫_{R_{θ,p}}^{γ} exp(−tp′s^{2σ}) (s − R_{θ,p})^{ℓ/2−1} ψ(s) ds]^{1/p′} ∀t ∈ [1, ∞),

where γ = (1+γ_{1} +γ_{2})^{1/4} and ψ is a continuous strictly positive function on the interval [R_{θ,p}, γ]. Now, Lemma 1 applies (where ω = 2σ, η = ℓ/2 and α = R_{θ,p}) and we conclude that

(3)  I(t) ≥ C t^{−ℓ/2p′} exp(−R_{θ,p}^{2σ} t)  ∀t ∈ [1, ∞);

combined with (2), this proves the lower bound of (iii).

We now prove the upper bound of (iii). From [3], Theorem 2.4, if p_{θ} ≤ p < 2, then

∥p_{t,θ}^{σ}∥_{p} ≤ ∥p_{t,θ}^{σ} ϕ_{iδ(p)ρ}∥_{1}^{δ(p)} N^{1−δ(p)},

where

N = [∫_{𝔞^{∗}} |p_{t,θ}^{σ} (Λ + iδ(p)ρ)|^{2} dμ̃(Λ)]^{1/2}

= [∫_{𝔞^{∗}} |exp(−t[Q_{θ}(Λ + iδ(p)ρ)]^{σ})|^{2} dμ̃(Λ)]^{1/2}

= [∫_{𝔞^{∗}} exp(−2t Re([Q_{θ}(Λ + iδ(p)ρ)]^{σ}))dμ̃(Λ)]^{1/2}.

Since p_{t,θ}^{σ} is a positive function,

∥p_{t,θ}^{σ} ϕ_{iδ(p)ρ}∥_{1} = p_{t,θ}^{σ} (−iδ(p)ρ) = exp(−R_{θ,p}^{2σ} t).

Thus

(4)  ∥p_{t,θ}^{σ}∥_{p} ≤ exp(−δ(p)R_{θ,p}^{2σ} t) N^{1−δ(p)}.

To estimate N, we observe that

Re([Q_{θ}(Λ + iδ(p)ρ)]^{σ}) ≥ (Re[Q_{θ}(Λ + iδ(p)ρ)])^{σ} = [⟨Λ, Λ⟩ + R_{θ,p}^{2} ]^{σ} ∀Λ ∈ 𝔞^{∗},

because, for any complex number z with nonnegative real part and any σ in (0, 1), Re(z^{σ}) ≥ (Re(z))^{σ}. We recall that dμ̃(Λ)/dΛ ≤ C(1 + |Λ|^{n−ℓ}), and pass to polar co-ordinates, to deduce that

N ≤ C[∫_{0}^{∞} exp(−2t[r^{2} + R_{θ,p}^{2} ]^{σ})(1 + r^{n−ℓ}) r^{ℓ−1} dr]^{1/2}
∼ [∫_{0}^{1} exp(−2t[r^{2} + R_{θ,p}^{2} ]^{σ}) r^{ℓ−1} dr

+ ∫_{1}^{∞} exp(−2t[r^{2} + R_{θ,p}^{2} ]^{σ}) r^{n−1} dr]^{1/2}
≤ C t^{−ℓ/4} exp(−R_{θ,p}^{2σ} t) ∀t ∈ [1, ∞),

by Lemma 2, so that

∥p_{t,θ}^{σ}∥_{p} ≤ exp(−δ(p)R_{θ,p}^{2σ} t) N^{1−δ(p)} ≤ C t^{−ℓ/2p′} exp(−R_{θ,p}^{2σ} t) ∀t ∈ [1, ∞), as required to prove (iii).

We now consider (iv). If p > 1 and ℓ > 1, then, much as argued to prove (2),

∥p_{t,θ}^{σ}∥_{p} ≥ C[∫_{𝔞^{∗}} |p_{t,θ}^{σ} (Λ + iδ(p)ρ)|^{p′} dμ̃(Λ)]^{1/p′}

≥ C[∫_{𝔠} exp(−tp′ |Q_{θ}(Λ + iδ(p)ρ)|^{σ}) dΛ]^{1/p′} ∀t ∈ [1, ∞),

where 𝔠 denotes the subset of 𝔞^{∗} of all elements of the form Λ_{0} + λ_{1}ρ, such that ⟨Λ_{0}, ρ⟩ = 0, ⟨Λ_{0}, Λ_{0}⟩ ≤ 1, and 0 ≤ λ_{1} ≤ 1. We denote the right hand side of the last inequality by J(t). Now if Λ may be written in this way, then

|Q_{θ}(Λ + iδ(p)ρ)| = |⟨Λ_{0}, Λ_{0}⟩ + λ_{1}^{2}R_{0}^{2} + 2iδ(p)λ_{1}R_{0}^{2}|
≤ ⟨Λ_{0}, Λ_{0}⟩ + λ_{1}^{2}R_{0}^{2} + 2λ_{1}R_{0}^{2}
≤ ⟨Λ_{0}, Λ_{0}⟩ + 3λ_{1}R_{0}^{2},

so, passing to polar co-ordinates in ρ^{⊥}, and then changing variables, we deduce that

J(t) ≥ C[∫_{0}^{1} ∫_{0}^{1} exp(−tp′ |λ_{0}^{2} + 3λ_{1}R_{0}^{2}|^{σ}) λ_{0}^{ℓ−2} dλ_{0} dλ_{1}]^{1/p′}

≥ C t^{−(ℓ+1)/2σp′}[∫_{0}^{t^{1/2σ}} ∫_{0}^{t^{1/σ}} exp(−p′ |λ_{0}^{2} + 3λ_{1}R_{0}^{2}|^{σ}) λ_{0}^{ℓ−2} dλ_{0} dλ_{1}]^{1/p′}

≥ C t^{−(ℓ+1)/2σp′} ∀t ∈ [1, ∞),

as required to prove the lower bound. If p = 1 or ℓ = 1, the argument simplifies but the conclusion is the same.

To prove the upper bound, we note first that (4) continues to hold, and that R_{θ,p}^{2σ} = 0, so that

∥p_{t,θ}^{σ}∥_{p} ≤ N^{1−δ(p)}.

To estimate N, we write Λ = Λ_{0} + λ_{1}ρ, where ⟨Λ_{0}, ρ⟩ = 0, as above, and note that

Re([Q_{θ}(Λ + iδ(p)ρ)]^{σ}) = Re([⟨Λ_{0}, Λ_{0}⟩ + λ_{1}^{2}R_{0}^{2} + 2iδ(p)λ_{1}R_{0}^{2}]^{σ})
≥ Re([⟨Λ_{0}, Λ_{0}⟩ + 2iδ(p)λ_{1}R_{0}^{2}]^{σ}).

We let Φ and Ψ : R^{+} × R → R be the functions given by the formulae

\begin{aligned}Φ(λ_{0}, λ_{1}) & = Re([λ_{0}^{2} + 2iδ(p)R_{0}^{2}λ_{1}]^{σ})  ∀λ_{0} ∈ (0, ∞), ∀λ_{1} ∈ (−∞, ∞) \\ Ψ(λ_{0}, λ_{1}) & = (1 + λ_{0} + |λ_{1}|)^{n−ℓ}  ∀λ_{0} ∈ (0, ∞), ∀λ_{1} ∈ (−∞, ∞), \end{aligned}

and recall that dμ̃(Λ)/dΛ ≤ C(1 + |Λ|^{n−ℓ}). By passing to polar co-ordinates in ρ^{⊥} and changing variables, we deduce that

N = [∫_{𝔞^{∗}} exp(−2t Re([Q_{θ}(Λ + iδ(p)ρ)]^{σ}))dμ̃(Λ)]^{1/2}

≤ C[∫_{−∞}^{∞} ∫_{0}^{∞} exp(−2t Φ(λ_{0}, λ_{1})) Ψ(λ_{0}, λ_{1}) λ_{0}^{ℓ−2} dλ_{0} dλ_{1}]^{1/2}

= C t^{−(ℓ+1)/4σ} [∫_{−∞}^{∞} ∫_{0}^{∞} exp(−2 Φ(λ_{0}, λ_{1})) Ψ(\frac{λ_{0}}{t^{1/2σ}} , \frac{λ_{1}}{t^{1/σ}}) λ_{0}^{ℓ−2} dλ_{0} dλ_{1}]^{1/2}

≤ C t^{−(ℓ+1)/4σ} ∀t ∈ [1, ∞).

Now we can conclude that

∥p_{t,θ}^{σ}∥_{p} ≤ C t^{−(1−δ(p))(ℓ+1)/4σ} = C t^{−(ℓ+1)/2σp′} ∀t ∈ [1, ∞),

as required. This finishes the proof of (iv), and of Lemma 3. ⊓ ⊔

For convenience, we list the results of our main theorem.

(i) if t > 0, then P_{t,θ}^{σ} is L^{p} – L^{q} -bounded only if p ≤ q, p ≤ p_{θ}′, and q ≥ p_{θ};

(ii) if p_{θ} ≤ p ≤ p_{θ}′, |||P_{t,θ}^{σ}|||_{p;p} = exp(−R_{θ,p}^{2σ} t) for all t in (0, ∞);

(iii) if p ≤ p_{θ}′ and q ≥ p_{θ}, |||P_{t,θ}^{σ}|||_{p;q} ∼ t^{−n(1/p−1/q)/2σ} for all t in (0, 1];

(iv) if p < q = 2 or 2 = p < q, |||P_{t,θ}^{σ}|||_{p;q} ∼ t^{−ν/4} exp(−R_{θ}^{2σ}t) for all t in [1, ∞);

(v) if p < 2 < q, |||P_{t,θ}^{σ}|||_{p;q} ∼ t^{−ν/2} exp(−R_{θ}^{2σ}t) for all t in [1, ∞);

(vi) if p < q < 2 and q > p_{θ}, |||P_{t,θ}^{σ}|||_{p;q} ∼ t^{−ℓ/2q′} exp(−R_{θ,q}^{2σ} t) for all t in [1, ∞);

(vii) if p < q = p_{θ}, |||P_{t,θ}^{σ}|||_{p;q} ∼ t^{−(ℓ+1)/2σq′} for all t in [1, ∞);

(viii) if 2 < p < q, |||P_{t,θ}^{σ}|||_{p;q} ∼ t^{−ℓ/2p} exp(−R_{θ,p}^{2σ} t) for all t in [1, ∞);

(ix) if p_{θ}′ = p < q, |||P_{t,θ}^{σ}|||_{p;q} ∼ t^{−(ℓ+1)/2σp} for all t in [1, ∞).

Proof of the theorem. First, a result of L. Hörmander [8] shows that P_{t,θ}^{σ} cannot be L^{p} –L^{q} -bounded unless p ≤ q. Next, P_{t,θ}^{σ} cannot be L^{p} –L^{q} -bounded when q < p_{θ}, by the sufficient condition of J. L. Clerc and E. M. Stein [1] and the fact that p_{t,θ}^{σ} continues analytically to T_{δ(p_{θ})} but to no larger tube. By duality, P_{t,θ}^{σ} cannot be L^{p} –L^{q} -bounded if p > p_{θ}′. This proves (i). Observe that parts (ii) to (ix) imply that P_{t,θ}^{σ} is L^{p} –L^{q} -bounded if p ≤ q, p ≤ p_{θ}′, and q ≥ p_{θ}.

We now prove (ii), for p in [p_{θ}, 2]. Define Λ_{p} to be iδ(p)ρ. By C. S. Herz’ principe de majoration [7] and spherical Fourier analysis, we have that

|||P_{t,θ}^{σ}|||_{p;p} = ∫_{G} p_{t,θ}^{σ}(x) ϕ_{Λ_{p}}(x) dx = p_{t,θ}^{σ} (−Λ_{p}) = exp(−R_{θ,p}^{2σ} t) ∀t ∈ (0, ∞).

By duality, this result also holds for p in [2, p_{θ}′].

Estimate (iii) is a consequence of the theory of ultracontractive semigroups, combined with the fact that |||P_{t,θ}^{σ}|||_{1;∞} = ∥p_{t,θ}^{σ}∥_{∞} and the estimate for ∥p_{t,θ}^{σ}∥_{∞} in Lemma 3 above. See, e.g., Cowling and Meda [4], E. B. Davies [5], or N. Th. Varopoulos et al. [9].

Next we prove (iv). By duality, it suffices to treat the case where p < 2 and q = 2. On the one hand, by the Kunze–Stein phenomenon [2], |||P_{t,θ}^{σ}|||_{p;2} ≤ C ∥p_{t,θ}^{σ}∥_{2}, so that

|||P_{t,θ}^{σ}|||_{p;2} ≤ C t^{−ν/4} exp(−R_{θ}^{2σ}t) ∀t ∈ [1, ∞),

from Lemma 3. On the other hand,

∥h_{1} ∗ p_{t,θ}^{σ}∥_{2} ≤ |||P_{t,θ}^{σ}|||_{p;2} ∥h_{1}∥_{p} = C |||P_{t,θ}^{σ}|||_{p;2} ∀t ∈ [1, ∞);

by the Plancherel formula, formula (1), and Lemma 2,

∥h_{1} ∗ p_{t,θ}^{σ}∥_{2} = [∫_{𝔞^{∗}} exp(−2tQ_{θ}(Λ)^{σ} − 2Q_{0}(Λ)) dμ(Λ)]^{1/2}

∼ [∫_{0}^{∞} exp(−2t[r^{2} + R_{θ}^{2}]^{σ} − 2[r^{2} + R_{0}^{2}]) (1 + r)^{n−ν} r^{ν−1} dr]^{1/2}

≥ C[∫_{0}^{1} exp(−2t[r^{2} + R_{θ}^{2}]^{σ}) r^{ν−1} dr]^{1/2}
∼ t^{−ν/4} exp(−R_{θ}^{2σ}t) ∀t ∈ [1, ∞),

so that

|||P_{t,θ}^{σ}|||_{p;2} ≥ C t^{−ν/4} exp(−R_{θ}^{2σ}t) ∀t ∈ [1, ∞).

It follows that |||P_{t,θ}^{σ}|||_{p;2} ∼ t^{−ν/4} exp(−R_{θ}^{2σ}t) for all t in [1, ∞), and (iv) is proved.

To prove (v), we proceed similarly. On the one hand, from (iv),

|||P_{t,θ}^{σ}|||_{p;q} ≤ |||P_{t/2,θ}^{σ}|||_{p;2} |||P_{t/2,θ}^{σ}|||_{2;q} ∼ t^{−ν/2} exp(−R_{θ}^{2σ}t) ∀t ∈ [1, ∞).

On the other hand,

|||H_{1,0}|||_{1;p} |||P_{t,θ}^{σ}|||_{p;q} |||H_{1,0}|||_{q;∞} ≥ ∥h_{2} ∗ p_{t,θ}^{σ}∥_{∞} ∼ t^{−ν/2} exp(−R_{θ}^{2σ}t) ∀t ∈ [1, ∞),

by an argument similar to that used in the proof of (iv) above. This completes the proof of (v).

We now prove (vi). On the one hand, Theorem 2.2 in [3] and Lemma 3 may be invoked to show that

|||P_{t,θ}^{σ}|||_{p;q} ≤ C ∥p_{t,θ}^{σ}∥_{q} ≤ C t^{−ℓ/2q′} exp(−R_{θ,q}^{2σ} t) ∀t ∈ [1, ∞).

On the other hand,

∥h_{1} ∗ p_{t,θ}^{σ}∥_{q} ≤ |||P_{t,θ}^{σ}|||_{p;q} ∥h_{1}∥_{p} = C |||P_{t,θ}^{σ}|||_{p;q} ∀t ∈ [1, ∞).

As argued to prove (2), if 𝔟 again denotes the unit ball in 𝔞^{∗},

∥h_{1} ∗ p_{t,θ}^{σ}∥_{q} ≥ C[∫_{𝔞^{∗}} |h̃_{1} (Λ + iδ(q)ρ) p_{t,θ}^{σ} (Λ + iδ(q)ρ)|^{q′} dμ̃(Λ)]^{1/q′}

≥ C[∫_{𝔟} |h̃_{1} (Λ + iδ(q)ρ) p_{t,θ}^{σ} (Λ + iδ(q)ρ)|^{q′} dΛ]^{1/q′}

≥ C[∫_{𝔟} |p_{t,θ}^{σ} (Λ + iδ(q)ρ)|^{q′} dΛ]^{1/q′} ∀t ∈ [1, ∞),

since for Λ in 𝔟, |h̃_{1} (Λ + iδ(q)ρ)| is bounded away from 0. This integral was treated in the proof of Lemma 3 (see (3)), and we conclude that

|||P_{t,θ}^{σ}|||_{p;q} ≥ C t^{−ℓ/2q′} exp(−R_{θ,q}^{2σ} t) ∀t ∈ [1, ∞),

completing the proof of (vi).

Finally, (vii) is proved in the same way as (vi), but part (iv) of Lemma 3 is used instead of part (iii), and (viii) and (ix) follow from (vi) and (vii) by duality. ⊓ ⊔

References

[1] Clerc, J. L., and E. M. Stein, L^{p} multipliers for noncompact symmetric spaces, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 3911–3912.

[2] Cowling, M. G., The Kunze–Stein phenomenon, Ann. of Math. 107 (1978), 209–234.

[3] Cowling, M. G., S. Giulini and S. Meda, L^{p} –L^{q} estimates for functions of the Laplace–Beltrami operator on noncompact symmetric spaces. I, Duke Math. J. 72 (1993), 109–150.

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

[5] Davies, E. B., “Heat Kernels and Spectral Theory,” Cambridge University Press, Cambridge, 1989.

[6]  Helgason, S., “Groups and Geometric Analysis,” Academic Press, 1984.

[7] Herz, C. S., Sur le phénomène de Kunze–Stein, C. R. Acad. Sci. Paris (Série A) 271 (1970), 491–493.

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

[9] Varopoulos, N. Th., L. Saloff–Coste, and T. Coulhon, “Analysis and Geometry on Groups,” Cambridge University Press, Cambridge, 1992.

M. G. Cowling School of Mathematics University of New South Wales, Sydney N.S.W. 2052 Australia e-mail: m.cowling@unsw.edu.au S. Giulini Dipartimento di Matematica Università di Genova via L. B. Alberti 4 I–16132 Genova, Italia e-mail: giulini@dima.unige.it

S. Meda

Dipartimento di Matematica

Politecnico di Milano

via Bonardi 9

I-20133 Milano, Italia

e-mail: stemed@ipmma1.mate.polimi.it

Received November 28, 1994