Pointwise and uniform bounds for functions of the Laplacian on non-compact symmetric spaces
Yulia Kuznetsova Zhipeng Song







1]Yulia Kuznetsova
[1]Université de Franche Comté, CNRS, LmB (UMR 6623), F-25000 Besançon, France


1,2]Zhipeng Song
[2]Ghent University, Department of Mathematics: Analysis, Logic and Discrete Mathematics, 9000 Ghent, Belgium




 Yulia Kuznetsova: yulia.kuznetsova@univ-fcomte.fr; Zhipeng Song: zhipeng.song@univ-fcomte.fr









[2020]22E30; 42B15; 35L05; 43A85; 43A90 




Pointwise and uniform bounds for functions of the Laplacian on non-compact symmetric spaces
    [
    September 9, 2024
===========================================================================================




Let L be the distinguished Laplacian on the Iwasawa AN group associated with a semisimple Lie group G. Assume F is a Borel function on ℝ^+. We give a condition on F such that the kernels of the functions F(L) are uniformly bounded. This condition involves the decay of F only and not its derivatives. 
By a known correspondence, this implies pointwise estimates for a wide range of functions of the Laplace-Beltrami operator on symmetric spaces. In particular, when G is of real rank one and F(x)= e^it√(x)ψ(√(x)), our bounds are sharp.






 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
































































































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§ INTRODUCTION


An important part of the classical harmonic analysis is the study of multipliers. Let k be a tempered distribution on ^n and m= k be its Fourier transform. The celebrated Hömander-Mikhlin multiplier theorem <cit.> asserts that the convolution operator T·:=· *k is bounded on L^p(^n) (1< p<∞) if the symbol m satisfies the Mikhlin type condition:

    sup_ξ∈^nξ^|α|∂_ξ^α m(ξ)≤ C< +∞,   ∀ |α|≤n/2+1.














Much of this theory makes sense on Lie groups, if we restrict our attention to spherical multipliers. Let G be a semisimple, connected, and noncompact Lie group. Assuming G has a finite center, it has an Iwasawa decomposition G=KAN, and there exists the Harish-Chandra transform , also called the spherical transform, taking K-biinvariant functions into functions on _^*. Here _^* is the dual of complexification _ of the Lie algebra  of A. Let k be a K-biinvariant tempered distribution on G and k its Harish-Chandra transform. Then the operator T·=· *k is a spherical multiplier on the symmetric space S=G/K, of kernel k and symbol m= k. 
We also refer reader to <cit.>.

The conditions on m are however very different from the Euclidean case: by a famous result of Clerc and Stein <cit.>, m must be holomorphic and bounded in an open tube around the real part ^* of _^*, in order to make T bounded on L^p(S) for 1<p<2. Conversely, if this condition is satisfied, then a Mikhlin-type bound completes the picture to give a sufficient condition. 
This was proved by Clerc and Stein for G complex, Stanton and Tomas <cit.>  in the rank-one case, and finally by Anker <cit.> in the case of a higher rank.

From the PDEs point of view, the most important class of multipliers is of those generated by the Laplace operator. We state the problem in the case of symmetric spaces directly. To the Riemannian structure on S, there corresponds canonically its Laplace–Beltrami operator Δ. It is self-adjoint on L^2(S). Consequently, by spectral theorem, every bounded Borel function F on  defines a bounded operator on L^2(S):

    F(Δ)=∫_0^∞ F(ξ) dE(ξ),
 
where E is the spectral measure of Δ. Moreover, being left-invariant, F(Δ) acts as a convolution on the right with a kernel k_F(Δ) (a priori defined in the distributional sense). 
An interesting question is, under what conditions on F, the operator F(Δ) is also bounded from L^p(S) to L^q(S)?
This requires estimating L^p norms of the kernel, which often requires finding its pointwise bounds. 

Such pointwise bounds are known for a variety of functions F and are obtained case by case. The heat kernel corresponding to exp(-t|Δ|) is very well studied, see the book <cit.> or a more recent survey <cit.>; the latest results <cit.> describe the asymptotics of solutions of the heat equation at large time. Moreover, bounds are available (most of them by Anker and coauthors) 
for:

 
 the resolvents (z-Δ)^-s <cit.>;
 
 the Poisson kernel corresponding to exp(-t√(|Δ|)) <cit.>;
 
 the Schrödinger kernel of the semigroup exp(itΔ) <cit.>;

 
 oscillating functions of the type (|Δ|+q)^-τ (|Δ|+q̃)^ σexp(it√(|Δ|+q)) with different exponents τ and σ, see <cit.> in rank one and recently <cit.> in higher ranks.




  * In rank one: Tataru <cit.>: the symbol is sin(λ t)λ (λ^2+β^2)^-ρ/2+is so that the operator is 
sin(√(|Δ_ρ|) t)√(|Δ_ρ|) (Δ_ρ+β^2)^-ρ/2+is, on the hyperbolic space (ℓ=1), the uniform bound is c(sinh t)^-ρ.

  * Anker, Pierfelice (on Damek-Ricci): D^-τD̃^τ-σ e^itD where D=√(|Δ|-ρ^2), D̃=√(|Δ|+ρ̃^2-ρ^2), ρ̃>ρ, σ∈_+, τ∈[0,3/2). That is, the operator is
 
    (|Δ|_ρ)^-τ/2 (|Δ|_ρ+ρ̃^2)^(τ-σ)/2 e^it√(|Δ|_ρ)

 <cit.>
 
  * Higher rank: Hassani <cit.>: |Δ|^-σ/2 e^it√(|Δ|); the bounds are for |t|≥1:

 |k_0(x)| ≤ C |t|^-l/2ϕ_0(x) (3+|x|)^a if a∈ 2 and l<a (integral over |λ|≤1);

 |k_∞(x)| ≤ C |t|^-dϕ_0(x) (3+|x|)^d if σ >n (integral over |λ|≥1).
 
  * Anker and Zhang <cit.>: |Δ|^-σ/2 e^it√(|Δ|), d=ℓ+∑_α∈Σ^+ m_α≥3 (but ℓ may be 1). Boundary point: σ = (d+1)/2. Since d=2n-l, this value is n-l/2+1/2.

 Theorem 3.7: for large t and |x|≥|t|/2, the bound is |t|^-N_1 (1+|x^+|)^N_2 e^-ρ(x^+) for every N_1∈ and N_2>N_1+2(d+1)+ (1/2) (max(d,D) -l).
 
 Theorem 3.3: For |x/t|<C_Σ<1/2, |ω̃^σ,0_t(x)| ≤ |t|^-D/2 (1+|x|)^(D-l)/2ϕ_0(x).
 



In this paper we obtain pointwise estimates for F(Δ) with a very general function F, on spaces of arbitrary rank. More precisely, we get estimates for the shifted Laplacian Δ_ρ = -Δ - |ρ|^2, which has no spectral gap; ρ is a fundamental linear functional on , see Section 2. Our main theorem (Corollary <ref> in the text) reads:

    Let F be a Borel function on ^+ such that 
    
    I_F =    ∫_^*F(λ^2) (1+|λ|)^7(n-l)+1 dλ < ∞.

    Then for every x∈ G
    
    |k_F(Δ_ρ)(x)| ≤ C I_F  e^-ρ(H(x)),

    where C is a constant depending on G only, and H(x) is the radial part of x in the Iwasawa decomposition.

See Section 2 for definition of the dimensions n, l.

This is equivalent to Theorem <ref> below. We show also that the upper limit of e^ρ k_F(Δ_ρ) at infinity is bounded by a similar integral with the power n-l instead of 7(n-l)+1.

It is clear that for F positive, one cannot hope for a much better bound: it is sufficient to evaluate k_F(Δ_ρ) at identity. 
But for oscillating functions (<ref>) can be close to optimal as well: we show in Theorem <ref> that for F(|λ|^2)=^it|λ|ψ(|λ|) and in rank one, the lower limit of e^ρ k_F(Δ_ρ) at infinity is never zero and is estimated from below by a similar integral. 

This generalizes results of <cit.> obtained for hyperbolic spaces.


From known estimates 
of spherical functions,




it is easy in fact to obtain bounds for the kernel of the type |k_F(Δ_ρ)(x)| ≲ (1+|H(x)|)^d e^-ρ(H(x)), see discussion at the end of Section 4. Our result is that it is possible to remove the polynomial factor in H(x).
To achieve this, we refine the asymptotic estimates of Harish-Chandra for spherical functions φ_λ. In the technical Section 3, we obtain explicit constants controlling the growth in λ in these estimates.


A polynomial factor in H(x) is often non-significant in the analysis of the Laplace-Beltrami operator. But it becomes essential in concern with the following operator L which has attracted much attention. 
We can define it as L = δ^-1/2τΔ_ρτδ^1/2, where τ is the inversion: (τ f)(g) = f(g^-1), and δ is the modular function of S. This makes L act as a convolution on the left, and the kernel of F(L) is linked to that of F(Δ_ρ) by k_F(L) = δ^-1/2 k_F(Δ_ρ). Importantly, L can be written as the sum of squares L=∑ X_j^2 of vector fields generating the Lie algebra of S, in a full analogy with the Euclidean case; whereas Δ has a necessary linear term in addition, due to non-unimodularity of S.







It turns out that the theory of L is very different from that of Δ: Hebisch <cit.> was first to prove that if G is complex, then F(L) is bounded on L^p(S) for 1≤ p<∞ as soon as  F satisfies the Mikhlin condition. In other words, no holomorphy is needed. 
In 1994 Cowling et al. <cit.> extended this result (with other constants replacing n/2+1) to all real Lie groups; they gave a sufficient condition on F such that F(L) is bounded on L^p for 1<p<∞ and is of weak type (1,1).
Sikora <cit.> obtained a stronger optimal order of differentiability, which is 1/2 smaller than the result of Cowling et al. Time-dependent bounds on L^1 for special oscillating functions were obtained by Gadziński <cit.>.

In the rank one case, estimates were obtained first by Müller and Thiele <cit.> on ax+b groups, and then by the same authors with Vallarino <cit.> on more general Damek-Ricci spaces, including all symmetric spaces of rank one. They show that for ψ ...., the kernel k_t of F(L) of type <ref> is bounded by k_t_1≲ (1+t) and k_t≲ 1. Akylzhanov, Kuznetsova, Ruzhansky, and Zhang <cit.> proved that on  ax+b groups, the estimates of Müller and Thiele are sharp.

Even earlier bu in an unpublished thesis Gadziński <cit.>  investigated  L_1- estimates of the function F(L) = e^it√(|L|) e^-bL with t,b>0. ...

 shorten this


We are interested in continuing this comparison. Thus, our Theorem <ref> is stated in terms of L: the kernel k_F(L) is uniformly bounded as soon as the integral (<ref>) converges.

We show finally that in rank one, this estimate is best possible even for oscillating functions of the type F(x) = exp(it√(x)) ψ(√(x)): the uniform norm of the kernel k_F(L) is bounded but does not decay at large time t. This generalizes the results of <cit.> for ax+b groups (more precisely, their parts concerning uniform norms); we note also a related subsequent result of Müller and Vallarino <cit.> on Damek-Ricci spaces, a class not covering all rank one symmetric spaces but containing many non-symmetric ones.





Acknowledgements. This work is supported by the EIPHI Graduate School (contract ANR-17-EURE-0002). The first author is also supported by the ANR-19-CE40-0002 grant of the French National Research Agency (ANR). The second author is also supported by the Methusalem programme of the Ghent University Special Research Fund (BOF), grant number 01M01021.




§ NOTATIONS AND PRELIMINARIES


Our main interest is in semisimple groups, but the proofs involve results on reductive groups. This class, also called the class of Harish-Chandra, is defined differently from one author to another. We stick to the following definition of Gangolli and Varadarajan <cit.>, and term it class ℋ accordingly:



A real Lie group G with its Lie algebra  is in class ℋ if

 is reductive;
    
     G has only finitely many connected components;
    	
     (G) ⊂ (_);
    
    and the analytic subgroup of G with Lie algebra , has finite center.
    



Every semisimple, connected Lie group with finite center is in class ℋ.




 §.§ Iwasawa decomposition

Let G be a Lie group in class ℋ with the Lie algebra .

Let K be a maximal compact subgroup of G and  the Lie algebra of K. We have the Cartan decomposition = ⊕ and the involution θ acting as 1 on  and -1 on . 
Let  be a maximal abelian subspace of , this is,  is a Lie subalgebra of  such that ⊂. 

We denote by ^* the real dual space of  and _^* the dual of its complexification _. 

Let Σ⊆^* be the set of restricted roots of (,). We have the restricted root space decomposition 

    = _0 ⊕⊕_α∈Σ_α

where _α={X∈|( H)X=α(H)X, ∀ H∈}. Setting =⊕_α∈Σ^+_α, we obtain the Iwasawa decomposition = ⊕⊕ with  abelian and  nilpotent.

Denote by m_α=_α the multiplicity of α. 
By choosing a lexicographic ordering of the roots we can define the set of positive roots Σ^+. Let Σ_r^+ be the set of reduced (also called short or indivisible) roots, that is, roots α∈Σ for which 1/2α is not a root. 
Let Σ_s^+⊂Σ^+ be the set of simple roots. 
Note that Σ_s^+ ⊆Σ_r^+ ⊆Σ^+ ⊆Σ; if G is semisimple, then Σ_s^+ is a basis of ^*.

Let A and N be the analytic subgroups of G with the Lie algebras  and . 



On the Lie group level, we get the Iwasawa decomposition G=KAN, and the multiplication map K × A × N → G given by (k, a, n)↦ kan is a diffeomorphism onto. 
Similarly, we also have the decomposition G=NAK. 
Let H(g) denote the unique -component of g∈ G in the decomposition g=kexp(H(g))n with k∈ K, n∈ N;
and let A(g) denote the unique -component in the decomposition g=nexp(A(g))k.

The decomposition (<ref>) is orthogonal with respect to the inner product defined on  as follows: 

    X,Y_θ := B_θ(X, Y)=-B(X,θ Y),  X,Y∈.

On × we shall write ·, · because B|_× = B_θ|_×.
The Killing form B on  is positive definite on . For every λ∈^*, there exists H_λ∈ such that λ(H)= H_λ, H for all H∈; this defines an inner product on ^*,

    λ, μ:= H_λ, H_μ,  λ, μ∈^*.

We denote in the sequel by |λ| the corresponding norm of λ∈^*.



Let  be the Weyl group of the pair (G, A). 
For any root α∈Σ, the operator

    s_α (φ)= φ-2φ, α/α^2α,  φ∈^*.

is a reflection in the hyperplane σ_α=φ∈^*|φ, α=0 and carries Σ to itself.
Now let us transfer s_α to  such that  

    s_α' (H)= H-2α(H)/α(H_α)H_α,  H∈.

Let  denote the finite group generated by all reflections s_α' (α∈Σ). 
For any s' ∈ and λ∈^*, we define s'λ:=λ^s'=λ∘ s'^-1 and then s_α(λ)= s_α ' λ. 
Let us consider the mapping u(k)=(k), k∈ M'. 
By the definition, the mapping (k)|_ : → only depends on the coset kM. 
Then u induces an homomorphism kM⟼(k)|_ of W into GL(). 
More precisely this is an isomorphism of W with ⊂ GL(). 
For convenience, we will use the same symbol sλ=s'λ to denote both reflections s'∈ and s generated by s_α. We also will use the same symbol  to denote W and . 


Set 

    _'=H∈_|α(H)≠ 0 for all α∈Σ

and '=∩_'. 


We define the positive Weyl chamber of  associated with Σ^+ as 

    ^+=H∈|α(H)> 0 for all α∈Σ^+.

We transfer this definition to _^*: 

    _^*'=μ∈_^*|α, μ≠ 0 for all α∈Σ,   ^*'=^* ∩_^*'.

Vectors in _^*' are said to be regular.

Let _+^* ⊆^* be the positive Weyl chamber associated with Σ^+, that is 

    _+^*=λ∈^* | λ, α>0,  for all α∈Σ^+.



We define the Weyl vector by

    ρ(H)=1/2tr( H|_) =1/2∑_α∈Σ^+m_αα(H)

for H∈. 
Denote by =H∈: α(H)≥ 0  for all α∈Σ^+ the closure of ^+ in . The polar decomposition of G is given by G=K K where =exp(), and we denote x^+ the -component of x∈ G in this decomposition.














 §.§ Spherical functions
 

Harmonic analysis on symmetric spaces is built upon spherical functions.
It is important to note that our main references, Helgason <cit.> and Gangolli and Varadarajan <cit.>, have different conventions in their notations. We chose to adopt Helgason's notations <cit.> in our work; translation from <cit.> to <cit.> is done by λ↦ iλ.


	Let f be a complex-valued function on G/K such that f̃(e)=1 and f∈ C^∞(G/K). f is called a spherical function if
	
		
  * f is K-invariant, that is, f∘τ(k^-1)=f for all k∈ K.
   τ is the conjugation?
		
  * f is a joint eigenfunction of all invariant differential operators, that is, for all D∈(G/K), Df=λ_Df where λ_D∈.  
	

For convenience, we say that f̃ is a spherical function on G if f is a spherical function on G/K. 
For each λ∈_^*, let φ_λ(x) be the elementary spherical function given by 

    φ_λ(x)= ∫_K e^(iλ-ρ)(H(xk)) dk   ∀ x∈ G.

Similarly, under the decomposition G=NAK we set

    φ_λ(x)= ∫_K e^(iλ+ρ)(A(kx)) dk   ∀ x∈ G.

These are exactly the joint eigenfunctions of all invariant differential operators on G/K.


We have the following functional equation: φ_λ= φ_sλ for all λ∈_^*, s∈.

In particular, the basic spherical function φ_0(x) =∫_K e^-ρ(H(xk)) dk, also denoted as Ξ(x), has the following estimates

    φ_0(exp H)≤ C(1+|H|)^d e^-ρ(H)  ∀ H∈^+.

then we have 

    ∫_B_r|φ_0(x)|^2dx ≤ C
    
    r^n    if r ≤ 1,
    
    r^ν   if r > 1.



More information on spherical functions is given by the Harish-Chandra -function. An explicit expression for it as a meromorphic function on _^* is given by the Gindikin-Karpelevich formula: For each λ∈_^*, 

    (λ) = c_0 ∏_α∈Σ_r^+ 2^-iλ, α_0Γ(iλ, α_0)/Γ(1/2 (1/2m_α +1+ iλ, α_0))
    Γ(1/2(1/2m_α +m_2α+ iλ, α_0)),

where α_0=α/α, α, Γ is the classical Γ-function, and c_0 is a constant given by (-iρ)=1. 
If we denote A(z, y)=Γ(z+y)/Γ(z), by Legendre duplication formula, the function  can be rewritten as

    (λ)^-1=C·∏_α∈Σ_r^+Aiλ, α_0, m_α/2 A1/2(iλ, α_0 +m_α/2) , m_2α/2.

From the formula above, one can deduce for λ∈^*: 


    |(λ)|^-2≤ C 
    
    |λ|^ν-l   if |λ|≤ 1,
    
    |λ|^n-l   if |λ|> 1.

where n:= G/K, l:= A, d is denoted by the cardinality of Σ_r^+, and ν:=2d+l which is called the 'pseudo-dimension'. 








This implies a less precise, but shorter estimate

    (λ)^-1≤
    C (1+|λ|)^1/2(n-l).

The other two properties of  we will use are that for λ∈^* and s∈

    |(sλ)|=|(λ)|,  (-λ)=(λ).


Let now Σ_s^+=α_1,⋯, α_r be the simple system in Σ^+, and Λ=^+α_1+⋯+^+ α_r the set of all linear combinations n_1α_1 + ⋯ + n_rα_r (n_i∈^+). Set Λ^+=Λ∖{0} and Λ̃=Λ=α_1+⋯+α_r. 
For any μ∈Λ̃ and s, t∈ (s≠ t) we define the following hyperplanes in _^*:

    σ_μ=λ∈_^*: μ, μ=2iμ,λ,


    τ_μ(s,t)=λ∈_^*:i(sλ-tλ)=μ.

Set σ=⋃_μ∈Λ^+σ_μ and σ^c=_^*\σ.
For μ=n_1α_1 + ⋯ + n_rα_r ∈Λ, we denote by 

    m(μ)=n_1 + ⋯ + n_r

the level of μ.

If λ∈_^* does not lie in any of the hyperplanes σ_μ or τ_ν(s,t), then φ_λ is decomposed into the Harish-Chandra series

    φ_λ(exp H)= ∑_s∈(s λ)^(isλ-ρ)(H)∑_μ∈ΛΓ_μ(sλ)^-μ(H),     H∈^+,

where Γ_μ are coefficient functions on ^* determined by the recurrent relation

    μ, μ-2iμ, λΓ_μ(λ)= 2∑_α∈Σ^+m_α∑_ k≥ 1,
    μ-2kα∈ΛΓ_μ-2kα(λ)μ+ρ-2kα,α-iα, λ,

with initial function Γ_0≡ 1. 

Let C(G//K) denote the space of continuous functions on G which are bi-invariant under K. 
Let C_c(G//K) be the set of all functions in C(G//K) with compact support. 

The spherical transform, called also the Harish-Chandra transform, of f∈ C_c(G//K) is defined by

    ( f)(λ)=∫_G f(x)φ_-λ(x)d x,   λ∈_^*.

The Harish-Chandra inversion formula is given by

    f(x)=C∫_^*( f)(λ)φ_λ(x)  (λ)^-2dλ,    x∈ G

where C is a constant associated with G. 




 §.§ Symmetric space



When G is a semisimple, connected Lie group with finite center, the homogeneous space G/K can be identified with the solvable Lie group S=AN with the Lie algebra ⊕. 
 The multiplication and inverse for (a,x),  (b,y)∈ S are given by
 
    (a,x) (b,y)=(ab,bxb^-1y)

 and 
 
    (a,x)^-1=(a^-1, ax^-1 a^-1),

which is well defined as  is stable under ().




The group G is unimodular, while S is not (unless G=A) and has the modular function given by 

    δ(s)=^-2ρ(log a),   s= (a,x)∈ S.

It admits an extension to G defined by δ(g):=^-2ρ(H(g))(g∈ G).
The left and right Haar measures of S are given by

    d_ls=d a d n,


    d_rs=δ^-1(s) d_ls=^2ρ(log a)d a d n.

As G is semisimple then is unimodular, the Haar measure d g on G is invariant and given by 

    d g=d_lkd_r(an)=d k d_r(an)=^2ρ(log a)d k d a d n.

or

    d g=d_l(an)d_rk=d_l(an)dk=d ad nd k.







 §.§ Distinguished Laplacian and Laplace-Beltrami operator

Keep the assumptions of Section <ref>.
The Killing form B of G is positive definite on . This defines a Riemannian structure on G/K and the Laplace–Beltrami operator Δ. It has also a coordinate description as described below.

The bilinear form B_θ: (X,Y)↦ -B(X,θ Y) is positive definite on  and the Iwasawa decomposition =⊕⊕ is orthogonal with respect to it. We choose and orthonormal basis (H_1,…,H_l,X_1,…,X_) in  so that (H_1,…,H_l) is a basis of  and (X_1,…,X_n-l) a basis of . With these viewed as left-invariant vector fields on , we have then <cit.>

    Δ = ∑_i=1^l H_i^2+2∑_j=1^n-l X_j^2+∑_i=1^l H_i · (H_iδ)(e).

For a function f on G, set τ (f) = f̌ and f̌(s):=f(s^-1). Setting -X = τ∘ X ∘τ for X∈, we get a right-invariant vector field; the operator L defined by

    -L = ∑_i=1^lH_i^2+2∑_j=1^n-lX_j^2

Denote by C_c^∞(S) the set of complex-valued continuous functions with compact support on S. 
Let  be the Lie algebra corresponding to S and left-invariant vector fields X_1,⋯, X_n be an orthogonal basis of . 
For each 1≤ i ≤ n, set X_i as the right-invariant vector fields on G that coincide with X_i at the identity which is denoted by e. 
One can have -X_i = τ∘ X_i ∘τ where τ (f) := f̌, for all f∈ C_c^∞(S) and f̌(s):=f(s^-1).

The inner products on C_c^∞(S) are defined by 

    f, g_l = ∫_S f(s)g(s)d_ls,    f, g_r = ∫_S f(s)g(s)d_rs

for f,g∈ C_c^∞(S). 
We define two operators L_r and L_l as follows

    L_r f, g_l =∑_j=1^nX_j f, X_j g_l,    L_l f, g_l =∑_j=1^n X_jf, X_jg_l

By <cit.> , we have
L_r=-∑_j=1^ nX_j^2 and L_l=-∑_j=1^ nX_j^2-∑_j=1^ n(X_jδ)(e)X_j. 
Particularly,
If we chose H_1, …, H_l to be an orthonormal basis of  and √(2)X_1, …, √(2)X_ to be an orthogonal basis of , define 

    -L_r = ∑_i=1^lH_i^2+2∑_j=1^ñX_j^2,


    -L_l= ∑_i=1^l H_i^2+2∑_j=1^ñ X_j^2+∑_i=1^l H_i · (H_iδ)(e).


One can prove that Δ acting on C^∞(G/K) can be identified with the operator -L_l. We call L_r the (minus) distinguished Laplacian on S and conveniently denote it by L.
is a distinguished Laplace operator on S=G/K and 
has a special relationship with Δ<cit.> :

    δ^-1/2∘τ L τ∘δ^1/2 = -Δ - ρ^2=: Δ_ρ.

The shifted operator Δ_ρ has the spectrum [0, ∞). Both Δ_ρ and L extend to positive and self-adjoint operators on L^2(S, d_l), with the same notations for their extensions.


Let F be a bounded Borel function on [0, ∞). 
We can define bounded operators F(L) and F(Δ_ρ) on L^2(S) via Borel functional calculus.  
Note that L is right-invariant while Δ_ρ is left-invariant, we can define k_ and k_ to be the kernels of F(Δ_ρ) and F(L) respectively, a priori distributions, such that 

    F(Δ_ρ)f=f *k_  and    
            F(L)f=k_*f,    f∈ L^2(S).

The connection between L and Δ implies that

    k_=δ^1/2 k_,

which was pointed out by Giulini and Mauceri <cit.> (also see <cit.>).
It is known that 
k_ is K-biinvariant. Thus, not only k_ but k_ can be regarded as a function on G by the formula above.
















By inverse formula for the spherical transform, we obtain the exact formula of the kernel as follows:

    k_(x)=C∫_^*F(|λ|^2)φ_λ(x)  (λ)^-2dλ,    x∈ G.






§ ASYMPTOTIC BEHAVIOR OF Φ_Λ

In this section, we shall introduce some notations and properties about the asymptotics of the elementary spherical functions. The main result is by Harish-Chandra <cit.>
, but we are systematically citing the book of Gangolli and Varadarajan <cit.>.


In this section, G is a group in class ℋ. We use all the notations introduced above.



Fix H_0∈ which is not in the center of . This is equivalent to say that the subset F⊂Σ_s^+ of simple roots vanishing at H_0 is not equal to Σ_s^+. Set _F = { H∈: (H)=0 for all ∈ F}, and let _1F be the centralizer of _F in . If we denote by M_1F the centralizer of _F in G, then _1F is corresponding Lie algebra of M_1F.

Denote by Σ_F^+ ⊂Σ^+ the set of positive roots vanishing at H_0 and Σ_F:=Σ_F^+∪ -Σ_F^+. Then F and Σ_F^+ are the simple and positive systems of the pair (_1F, ) respectively.
With =Z_(), we have the restricted root space decomposition

    _1F=⊕⊕⊕_λ∈Σ_F_λ,

where as expected 

    _λ = X∈_1F: H,X=λ(H)X for all H∈
 



for each λ∈Σ_F. 

Let _F=_1F∩ 
and _1F=_1F∩=⊕_λ∈Σ_F^+_λ. 
Then we have 
the Iwasawa decomposition _1F=_F⊕⊕_1F. 
Let K_F=K∩ M_1F and N_1F=exp_1F, on Lie group level, we also have the decomposition 
M_1F=K_F A N_1F.
We define θ_F=θ|_M_1F and B_F=B|__1F_×_1F_, then θ_F is a Cartan involution of M_1F. It is important that M_1F is also of class ℋ and (M_1F,K_F,θ_F,B_F) inherit all the properties of (G,K,θ,B)<cit.>.

We denote the Weyl group of pair (_1F,) by _F and define τ_F(H)=min_α∈Σ^+\Σ_F^+α(H) for all H∈. 
Let _F denote the Harish-Chandra -function with respect to M_1F. 
The elementary spherical functions on M_1F corresponding to λ∈_^* are given by

    θ_λ(m)=∫_K_F^(iλ-ρ_F)(H(mk_F))dk_F,

where ρ_F:=1/2∑_α∈Σ_F^+m_αα. 

An important part of our proofs relies on Theorem <ref> below, which is <cit.>. 
To state it, we need the following result. It is a proposition in <cit.> since the functions ψ_λ on M_1F are defined there by another formula <cit.>; we take it however for a definition.

[<cit.>]
	Let λ∈^* be regular. Then 
	
    ψ_λ(m)=_F^-1∑_s∈ ((sλ)/_F(sλ))θ_sλ(m)

	for all m∈ M_1F.
 


Next, we introduce the following theorem about the asymptotic behavior of φ_λ in a region including a wall of the positive chamber. 
For any ζ> 0, define 

    A^+(H_0:ζ):=a∈ : τ_F(log a) > ζ|log a| .

This is a conical open set in , and it contains exp H_0 when ζ is small enough.







Fix ζ >0. Then we can find constants κ=κ(ζ)>0, 

C=C(ζ)>0, and ι=ι(ζ)>0 such that for all λ∈^* and all a∈ A^+(H_0: ζ),

    |^ρ(log a)φ_λ(a)-^ρ_F(log a)ψ_λ(a)| 
    ≤ C(1+λ)^ι^-κlog a.





 §.§ Estimates of the constants in Theorem <ref>


We are interested in explicit values of the constants in the theorem above, and especially of ι. Theorem <ref> as stated is in fact a particular case of <cit.>: the latter contains estimates for a differential operator b acting on φ_λ, and we apply it in Section <ref> with b=𝕀. We would like to estimate the constant ι, which is denoted by s in the book <cit.>, in the case of a general operator b, since the proof is almost the same.


Extracting constant from the proofs of <cit.> and its preceding statements is a long task requiring a lot of citations. 
It is impossible to explain every symbol appearing in the discussion and to keep a reasonable volume, so we invite the reader to consult the book <cit.> in parallel. Triple-numbered citations like Proposition 5.8.3 or (5.9.2) will be always from this book.  Any new notations introduced in this subsection will be used only within it.


First to mention are 

 the notations concerning parabolic subgroups. Those introduced above suit our proofs in Lemma <ref>, but are different from <cit.>. In this subsection we denote, following the book <cit.>: M_10=M_1F, _10=_1F, _0=_F, τ_0=τ_F and ρ_0=ρ_F.

Let _10 denote the algebra _10∩ (we note that in <cit.>, the complement of  in the Cartan decomposition is denoted by  rather than ).
Denote by U() and U(_10) the universal enveloping algebras of  and _10 respectively. There is a projection β_0 of U() onto U(_10) defined in (5.3.32) and (5.3.33).

Next, a map γ_0 is defined in (5.9.2) by

    γ_0(b) = d_P_0∘β_0(b) ∘ d_P_0^-1,   b∈ U()

where d_P_0 is a homomorphism from M_10 to _+ defined on p. 208 referring to the formula (2.4.8). For a∈ A, we have d_P_0(a) = e^(ρ-ρ_0)(log a) by Lemma 5.6.1. 

The space ^*_ is denoted by ℱ in <cit.>, and is decomposed as ℱ = ℱ_R⊕ℱ_I with ℱ_R=^* and ℱ_I = i^*. For λ∈ℱ this gives λ=λ_ R+λ_ I.
For any κ>0, the set ℱ_I(κ) is defined in Lemma 4.3.3 as

    ℱ_I(κ) = {λ∈ℱ: |λ_ R| <κ}.

In particular, it contains ℱ_I. 




It is important to recall again that a spherical function denoted by φ_λ in <cit.> would be denoted φ_iλ in <cit.>, so that φ_λ of <cit.> is positive definite when λ∈ℱ_I (Proposition 3.1.4), and the integral in the inversion formula of the spherical transform is over ℱ_I as well (Theorem 6.4.1). Thus, in Proposition <ref> below we consider the case λ∈ℱ_I which implies λ_ R=0; in Corollary <ref>, we return to the notations of the rest of the paper.

We can now state the theorem:

Theorem 3.2' <cit.>





Fix ζ >0. Then we can find constants κ=κ(ζ)>0, and, for each b∈ U(), constants 
C=C(ζ,b)>0, s=s(ζ,b)>0 such that for all λ∈ℱ_I(κ) and all a∈ A^+(H_0: ζ),

    |^ρ(log a)(bφ_λ)(a)-^ρ_0(log a) (γ_0(b) ψ_λ)(a)| 
    ≤ C(1+λ)^s^-κlog a.



Our result is now:


For regular λ∈ℱ_I, the inequality (<ref>) holds with the constant s = 6d +1 + b where d=Σ_r^+ is the number of indivisible roots.


As we will see, this constant is determined by the degrees of differential operators involved, so we will follow the proofs very briefly while keeping a particular attention at this point.

Two more remarks on notations are also needed. If μ is a differential operator and f a function (on G, or its subgroup), then we denote its value at a point x by (μ f)(x), but in <cit.> it is denoted also by f(x;μ). Also, the spherical functions in <cit.> are denoted both φ_λ(x) and φ(λ:x).


The proof of Theorem 5.9.3(b) relies on Proposition 5.9.2(b) with the same power s.

In its proof in turn, estimates with (1+|λ|)^s come from two sides:
 

    
  * Formula (5.8.8) of Proposition 5.8.3 estimating |Φ_0(λ:mexp H;μ) - Θ (λ:mexp H;μ)|, with μ=γ_0(b).





    The function Φ_0 is defined in (5.4.12), and Θ in Proposition 5.8.1. Formula (5.8.8) is a vector-valued inequality: by (5.4.12), for λ∈^*_, m∈ M_10
    
    Φ_0(λ:m) = [ (v_1∘ d_P_0) φ_λ(m);                   ⋮; (v_k∘ d_P_0) φ_λ(m) ]

    with operators v_1, …, v_k∈ U(_10) defined in (5.4.9). In Proposition 5.9.2, only the first coordinate is used, in which v_1=1 by (5.4.5).

  * The estimates of (μ_i φ_λ)(m exp H) with m∈ M_10, H∈, which are obtained by Lemma 5.6.3 with μ=d_P_0∘μ_i ∘ d_P_0^-1; here μ_i, 1≤ i≤ N, are operators in U(_10) whose existence follows from Proposition 5.3.10 with g=b (since φ_λ is K-biinvariant, we have (b φ_λ)(m)=(δ'(b)φ_λ)(m) by (5.3.26), at m∈ M^+_10A_+ as in the proof of Proposition 5.9.2). We get also from Proposition 5.3.10 that μ_i≤ b for all i.


Discussion of the case (b).
Case (a) will require a longer discussion, but it is eventually reduced to the same Lemma 5.6.3, so we continue first with the case (b).
Let us state the lemma in question









<cit.>: Denote by U(_10) the universal enveloping algebra of _10.
    Fix μ∈ U(_10). Then there are constants C=C(μ)>0, s=s(μ)≥ 0 such that
    
    | (μ∘ d_P_0) φ_λ(m) | ≤ C e^|λ_ R| σ(m)  (1+|λ|)^s (1+σ(m))^s Ξ_0(m)

    for all λ∈_^*, m∈ M_10.


The function σ is defined in (4.6.24), but in fact we do not need to consider it since this factor does not depend on λ. We recall again that notations of <cit.> are different from that of Helgason and of those used in our paper, so that for λ∈^*⊂^*_ we have λ_ R=0 in the cited Lemma.

In the proof of Lemma 5.6.3, the constant s comes from the formula (4.6.5) in Proposition 4.6.2:

    |φ(λ; ∂(v):x; u)| ≤ C (1+|λ|)^ u(1+h(x))^ vφ(λ_ R:x).

When applied in Lemma 5.6.3, one puts v=1 so that ∂(v)=𝕀, then x=m and u=d_P_0^-1∘μ∘ d_P_0, so that u = μ. The function h is defined in the same Proposition 4.6.2 but it is not significant in our case since v=0.

We see now that the constant s appearing in Lemma 5.6.3 is μ, and by our previous considerations, it is bounded by b in the case (b).

Discussion of the case (a).
In the proof of Proposition 5.8.3, the estimate (5.8.8) follows from (5.8.16), where one can replace exp(Γ_0(λ:H)) Θ^μ(λ:m) by Θ(λ:mexp H;μ) according to (5.8.18) and (5.8.20).
And (5.8.16) is an application of Proposition 5.7.4, precisely (5.7.26). One should note that (5.7.26) is an inequality involving functions of t∈^q, whereas in (5.8.16) this variable is hidden: we suppose that H=∑_i=1^q t_i H_i, where H_1,…,H_q are chosen on p.224.

The factor (1+|λ|)^s in (5.8.16) is a product of two others. The first one comes from the factor (1+Γ)^2(k-1) in (5.7.26) since Γ_i=Γ_0(λ:H_i), with Γ_0 defined in (5.5.18). For every i, the norm of the matrix Γ_0(λ:H_i) is bounded by |λ| |H_i| since its eigenvalues are of the form H_i(s^-1λ), s∈, by Theorem 5.5.13. Moreover, k in this formula is the dimension of the space of values of the function f; in (5.8.16), the function under study is Φ_0 (with λ, m, μ fixed), and its values are k-dimensional with k as in (5.4.12), that is, k=[:_0], by (5.4.4).

The second factor contributing to (1+|λ|)^s in (5.8.16) is G_i,λ,m_β |λ_ R| ,s estimated in (5.8.9), with the function G_i,λ,m defined in (5.6.4), and the power s is given by Lemma 5.6.6; recall that this lemma is applied with μ=γ_0(b). Note also that (5.8.9) does not add an exponential factor in λ since λ_ R=0 for λ∈^*.
    
Altogether, we get (1+|λ|) in the power 2([:_0]-1)+s, where s is given by Lemma 5.6.6 (second inequality) with μ=γ_0(b).
The index [:_0] can however be large and is not satisfactory for us. But actually it replaced by a better estimate, and let us see how.

Improving the estimate (1+Γ)^2(k-1).
We have seen that this factor comes from (5.7.26), with Γ_i = Γ_0(λ:H_i), i=1,…,q. Formula (5.7.26) should read

    | f(t) - exp(t_1 Γ_1+…+t_q Γ_q) θ| ≤ C (1+Γ)^2(k-1) (1+| t|)^s+2k-1max_i  g _i_b,s e^-(aη/3)| t|

(there is a misprint in the book where the exponent is exp(-t_1 Γ_1+…+t_q Γ_q); the first minus sign is unnecessary, as can be figured out from the proof, or compared with (5.7.14); when applied in Proposition 5.8.3, this minus is not supposed either).
This inequality is obtained by estimating first | f^*( t)-θ| with f^*( t) = exp(-t_1 Γ_1-…-t_q Γ_q)  f( t) (notations of (5.7.15)) then multiplying by exp(t_1 Γ_1+…+t_q Γ_q). In our case, 

    t_1 Γ_1+…+t_q Γ_q = t_1 Γ_0(λ:H_1)+…+t_q Γ_0(λ:H_q) = Γ_0(λ:H),

since Γ_0 is linear in H_i (see (5.5.16)-(5.5.19)). Let us write L=Γ_0(λ:H) till the end of this proof. Thus, we are estimating

    | f(t) - exp(L) θ| ≤exp(L) | f^*(t)-θ|.

The estimate for | f^*(t)-θ| is obtained as | f^*(t)- f^*(t,…,t)| + | f^*(t,…,t)-θ| where t=min( t), and both are eventually reduced to Corollary 5.7.2 of Lemma 5.7.1. We would like to replace it by the following:

Claim. If λ∈^* is regular, then exp(L)≤ C_k (1+|λ|)^2ξ for every H
∈, where ξ = max_1≤ j≤ k u_j and u_j are defined in (5.5.14).

    Let  denote the coset space /_0. By Theorem 5.5.13, L=Γ_0(λ:H) is diagonalizable: for s̅∈, the vector f_s̅(λ) of (5.5.34) is its eigenvector with the eigenvalue H(s^-1λ), and these vectors form a basis of ^k (recall that k=||). The matrix F(λ) with columns f_s̅(λ) is the transition matrix to the Jordan form J(λ) of L=F(λ) J(λ) F(λ)^-1, so that
    exp L=F(λ) exp( J(λ) ) F(λ)^-1. We have now
    
    exp L≤ F(λ)  exp( J(λ) )  F(λ)^-1≤F(λ)  F(λ)^-1

    since J(λ) is diagonal with purely imaginary values on the diagonal (recall that we assume λ∈ 
    i ^*).
    
    The norm of F(λ) is up to a constant (depending on k) equal to the maximum of its coordinates, that is, of u_j(s^-1λ) over 1≤ j≤ k and s∈, by Theorem 5.5.13. Every u_j is a polynomial function on ^* of degree u_j, chosen in (5.5.14). We can thus estimate F(λ)≤ C_k max_j (1+|λ|)^ u_j.
    
    The inverse matrix has, by Lemma 5.5.10, coordinates |_0|  u^i(s̅^-1λ) where (u^i) is a basis dual to (u_j) with respect to the bilinear form (5.5.25). Both bases span the same linear space. Belonging to the linear span of (u_j), every u^i has degree at most max_j  u_j, and this allows to estimate F(λ)^-1 by C_k max_j (1+|λ|)^ u_j and finally
    exp L≤ C_k max_j (1+|λ|)^2 u_j, what proves the claim.


Return now to the proof of (5.7.26). We can use now the Claim above instead of every reference to Lemma 5.7.1 or Corollary 5.7.2. One occurrence is (5.7.27), but let us first recall the notations (5.7.7)-(5.7.8) which imply that for any continuous ^k-valued function h on ^q_+ and any t∈^q_+,

    | h(t)| ≤ k^1/2 e^b| t| - amin ( t)  (1+| t|)^s  h_b,s.

Note that | t| = |t_1|+…+|t_q| and min( t)=min (t_1,…, t_q) by (5.6.6). In particular,

    | h((t,…, t))| ≤ k^1/2 e^bqt-at (1+ qt)^s  h_b,s.

This together with the Claim implies that (5.7.27) can be improved to


    ∑_1≤ i≤ q | g_i^*(t,…,t)| ≤ C  (1+qt)^s (1+λ)^2ξ  e^qbt-atmax_1≤ i≤ q g_i_b,s.

By (5.7.23), qb-a<aη/3-a<-2a/3. The existence of the limit θ follows as in the original proof, and this limit is given by (5.7.12). Up to a change of notations, this is also (5.7.20). We can now estimate

    | f^*(t,…,t)-θ| ≤ C max_1≤ i≤ q g_i_b,s (1+|λ|)^2ξ∫_t^∞ (1+qu)^s e^(qb-a)u du,

with the integral

    ∫_0^∞ (1+qt+qv)^s e^(qb-a)(t+v) dv ≤ C_a,b,s (1+t)^s e^(qb-a)t.

Next comes the estimate of | f^*( t)- f^*(t,…,t)|, done similarly to the calculations between (5.7.21) and (5.7.22). We arrive at

    | f^*( t)- f^*(t,…,t)| ≤ C max_1≤ i≤ q g_i_b,s (1+|λ|)^2ξ (1+| t|)^s e^b| t|-a min( t)

instead of (5.7.22), and conclude that for t∈ S_η (this set is defined in (5.7.4), and within it t≥η| t|)

    | f( t) - exp(L) θ| ≤ (1+|λ|)^4ξmax_1≤ i≤ q g_i_b,s (1+t)^s e^(b-η)| t|,

allowing to replace (1+Γ)^2(k-1) by (1+|λ|)^4ξ in (5.7.26).

Collecting the powers of (1+|λ|). The final power of (1+|λ|) is the maximum between two cases: in case (b), s≤ b; in case (a), s is bounded by the sum of 4ξ = 4max_j  u_j with u_j of (5.5.14 ), and of the power s given by Lemma 5.6.6 (second inequality) with μ=γ_0(b), of degree μ≤ b. We will proceed now to estimate this last power.

Lemma 5.6.6 with its two inequalities follows directly from Lemma 5.6.4, first and second inequalities respectively. The power s and the operator μ are the same in both. And the second inequality of Lemma 5.6.4 is reduced to the first one, with another μ; we need to estimate its degree. For a given i, the operator E_i=E_H_i is defined in (5.4.14) as a k× k matrix with nonzero entries(only in the first column) of the type

    ∑_p=1^p_iψ_H_i,jpμ_H_i,jp,

j=1,…,k, where ψ_H_i,jp are functions in R^+ (that is, smooth functions on A^+, see a remark after formula (5.2.7) and notations after (4.1.26)), and μ_H_i,jp are differential operators in U(_10) of degree

    μ_H_i:jp≤ H_i +  v_j = 1 +  v_j,

by (5.4.13).
The operators v_j are introduced by the formula (5.4.9), and we will yet return to them. The product μ E_i is a matrix again with entries(only in the first column) 

    ∑_p=1^p_iμ∘ (ψ_H_i,jpμ_H_i,jp) = ∑_p=1^p_i∑_l=1^l_iψ_ijpl' μ'_ijplμ_H_i,jp,

j=1,…, k, with some ψ_ijpl'∈ R^+, μ'_ijpl∈ U(_10), and each entry in (<ref>) has degree ≤μ+1+max_j  v_j. This is thus the maximal degree of μ appearing in the first inequality of Lemma 5.6.4. The last reduction to make is to use the formula (5.4.12) defining Φ_0, also cited above; estimates for Φ_0(λ:m;μ) follow from those for φ(λ:m; μ∘ d_P_0) in Lemma 5.6.3, but with μ∘ v_j, j=1,…,k, instead of μ. This increases the bound for the degree in (<ref>) to

    μ+1+2max_j  v_j

with (v_j) of (5.4.9). Now, as we have seen in the discussion of the case (b) above, the constant s in Lemma 5.6.3 is μ, so that s= s(μ) ≤ b+1+2max_j  v_j with μ=γ_0(b).

The total estimate for s in the initial Theorem 5.9.3 is now

    s ≤ b+1+2max_j  v_j + 4 max_j  u_j

with (u_j) of (5.5.14) and (v_j) of (5.4.9).




p.243: d_P_0(a) = e^(ρ-ρ_F)(a).

: for m∈ M_10

    d_P_0(m) = |  Ad(m) |__0| ^1/2.

Here _0 is the nilpotent component in the Langlands decomposition of P_0=M_0A_0N_0; in our case P_0=P_F is standard and _F = ∑_λ∈Δ^+∖Δ^+_F_λ (2.3.15); by (2.4.10),

    d_P_0(ma) = exp( 1/2 Ad (log a) |__0),

and we can write explicitly Ad (H) |__0 = ∑_λ∈Δ^+∖Δ^+_F m_λλ(H) for H∈.

We are at our final destination: s is u, and we can trace it back.
In Lemma 5.6.3 we have u=d_P_0^-1∘μ∘ d_P_0 with μ∈ U__10, so that u = μ.

When used in Proposition 5.8.3, formula (5.8.9) estimates G_i,λ,m_βλ_ R, s. Notations of the cited Lemma 5.6.6 are a bit different: 

    | G_i (λ: m: t) |    ≤ C e^λ_ Rσ(m) + βλ_ R |t|  (1+λ)^s
     
        (1+|t|)^s (1+σ(m))^s Ξ_0(m) ε_0(m)^2 ( 1-ε_0(m)^2 )^-(s+1) e^-2min(t);

the norm G_i,λ,m_βλ_ R, s as a function of t∈^q is defined by (5.7.7) (see the proof of Proposition 5.8.3).


Now, the second formula in Lemma 5.6.4 which we apply with μ=γ_0(b) reduces to the first one

    | Φ_0(λ:m;μ) | ≤ C e^λ_ Rσ(m) (1+λ)^s (1+σ(m))^s Ξ_0(m)

with μ=γ_0(b)ψ_H_i:jpμ_H_i:jp. Together with the definition of Φ_0 by (5.4.12)

    Φ_0(λ:m) = [ φ(λ:m; v_1∘ d_P_0);                  …; φ(λ:m; v_k∘ d_P_0) ]

this makes us apply Lemma 5.6.3, that is, (<ref>), with μ = γ_0(b)ψ_H_i:jpμ_H_i:jp v_k which is of degree ≤ 1 +  b + 2max{ v_j: 1≤ j≤ k}.





Concentrate now on the degrees of v_j. These operators are introduced by the formula (5.4.9): v_j∈𝔔_0=U(𝔰_10)^K_0 are such that γ__10/(v_j)=u_j; the elements u_j∈ U() are homogeneous and _0-invariant. The homomorphism γ__10/ is defined by the formula (2.6.53) and Theorem 2.6.7. By the proof of Theorem 2.6.7 (p. 93-94) v_j can be chosen of the same degree as u_j.

Now by (5.4.1-5.4.5) u_j are in ℌ, a graded subspace of U() such that U() is the direct sum of ℌ and U()I_^+, where I_ is the space of -invariant polynomials in U() and I_^+ is the subset of polynomials in I_ of positive degree. This situation is described by general theory in <cit.>. In particular, <cit.> states that the multiplication map I_⊗ℌ→ U() is an isomorphism, and the Poincaré series of ℌ is given by

    P_ℌ(t) = ∏_1≤ i≤ l (1+t+… +t^d_i-1),

where d_i= p_i are the degrees of polynomials generating I_. There are exactly l of them, see <cit.>; moreover, by (4.15.34) we know that ∑_i=1^l (d_i-1) is the number of reflections in . This is exactly the number of indivisible roots in Δ^+ (see for example Corollary 4.15.16, and note that s_α = s_2α if both are roots). This gives us an estimate

    ∑_i=1^l (d_i-1) ≤ d.

By (<ref>), the left hand side is also the highest degree of polynomials in ℌ, whence we conclude that v_j≤ d.

The reasoning for u_j is similar, just done in a more abstract setting; the bound is d as well. 

We obtain finally:

    s 
    ≤ b + 6d + 1.




In Theorem <ref> we can choose ι≤ 6d+1 ≤ 6(n-l)+1.


This is just <cit.> applied with b=𝕀 and observe that Σ_r^+≤Σ^+≤ = n-l.



    We would like to say a bit more about the index k=[:_0]. In terms of d (the number of indivisible roots), it can grow exponentially: if _0={𝕀} then k=|| and is, for example, (d+1)! if the root system is A_d. In our applications _0 will be the parabolic subgroup generated by all simple roots but one; however, in this case k can also grow exponentially. An example is given again by the system A_d. Suppose that d=2p is even and remove the p-th root; the remaining system is then reducible and _0 is the direct product of A_p and A_p-1. We have k =  (2p+1)! (p+1)!  p!, and up to a constant this is 4^p/√(p). Our final estimate by 6d+1 is thus indeed much lower.

    [Calculations are to be removed in the published version:

    k =  (2p+1)!/ (p+1)!  p!    ∼√(2π (2p+1) ) (2p+1)^2p+1 e^p+1 e^p / e^2p+1√(2π (p+1) 2π p) (p+1)^p+1 p^p 
       ∼√( 2p+1 ) (2p+1)^2p+1/√(2π (p+1) p) (p+1)^p+1 p^p 
       ∼ (2p+1)^2p+1/√(π p)  (p+1)^p+1 p^p 
       ∼ 2^2p+1( p+1/2/p+1) ^p+1( p+1/2/p) ^p (π p)^-1/2
       ∼ 2^2p+1exp( (p+1) ln ( 1+ 1/(2p+2) ) ) exp( p ln(1+1/(2p)) ) (π p)^-1/2
       ∼ 2^2p+1exp( (p+1)/(2p+2) ) ) exp( p /(2p)) ) (π p)^-1/2
       ∼ 2^2p+1 e^1/2+1/2 (π p)^-1/2
       ∼ 2^2p+1 e (π p)^-1/2
    
]





Estimates of κ.
In Lemma 5.6.6, β is defined as max_1≤ i≤ q |H_i|, where (H_i) is a basis of _0∩, which extends to a dual basis of  with respect to the simple roots (α_i).
In the proof of Proposition 5.8.3, after the formula (5.8.15), the constants c and η>0 are chosen so that ^+_0(η) is nonempty, 0<c<1 and for every t∈^q

    |t_1 H_1+… +t_q H_q| ≥ ct.

Then one sets

    κ =  2cη/3q β (2k+1).

All subsequent results relying on this Proposition use in fact the same value of κ.

Let us analyse it.
By (5.6.3), ^+_0 = { H∈_0: α_i(H)>0, 1≤ i≤ q}, and by (5.8.1),

    _0^+(η) =  { H∈_0: τ_0(H) > η|H| },

where τ_0(H) = min_1≤ i≤ q |α_i(H)| by (5.6.3).

It follows that κ depends on the root system only. In other words, for a given group it can be chosen once and be valid for any choice of H_0 or ζ.













§ UNIFORM ESTIMATES OF KERNELS


This is the central part of our article. We start with a general lemma concerning functions on a general group G in class ℋ, and then we derive from it estimates for the kernels of a class of functions of Laplace operators, for G semisimple.





    Let G be a group of class ℋ and R a radial positive function on ^*. 
    Then 
    
    sup_a∈^ρ (log a)∫_^*R(λ)φ_λ(a)(λ)^-1d λ≤ C  ∫_^* R(λ) (1+|λ|)^6d+n-l/2+1 dλ,

    where the constant C>0 depends on the group only.
Moreover,

    lim sup_a∈ |log a|→∞^ρ (log a)∫_^*R(λ)φ_λ(a)(λ)^-1d λ≤ C ∫_^* R(λ) (1+|λ|)^n-l/2 dλ.




It will be clear from the proof that one can replace the estimates by

    C ∫_^* R(λ) (1+|λ|)^n-l+1 |(λ)|^-1 dλ   and    
    C ∫_^* R(λ) (1+|λ|)^n-l/2 |(λ)|^-1 dλ

respectively, which is more precise around 0 (in ^*).



	
Step 1.

First, we deal with the simplest case Σ_s^+=1. 
Throughout this step we assume that Σ_s^+=α. Then Σ^+=α or Σ^+=α,2α, and Σ=±α or Σ=±α,± 2α. 
Since the argument will be similar and easier when Σ^+=α, in the following proof, we only discuss the case Σ^+=α,2α. 
Under the assumption above we have

    ^+=H∈: α(H)> 0,  =H∈: α(H)≥ 0.


With the fact s_α=s_-α=s_2α we know that =𝕀, s_α and Λ=α^+. We rewrite the Harish-Chandra series of φ_λ as follows: for all regular λ∈^* and all H∈^+,

    φ_λ(exp H)^ρ(H)
    = (λ)^iλ(H)∑_m=0^∞Γ_mα(λ) ^-mα(H)
    + (s_αλ)^i s_αλ(H)∑_m=0^∞Γ_mα(s_αλ) ^-mα(H).

By <cit.>, there exist constants C>0, p≥ 
0 such that |Γ_mα(λ)| ≤ C m^p for all λ∈^*.

We now need an estimate for Γ_mα. By definition, Γ_0≡1 and for each m≥1

    m^2α^2-2miα, λΓ_mα(λ)
    	   = 2m_α∑_k=1^m/2Γ_m-2kα(λ) mα+ρ-2kα,α-iα, λ
          + 2m_2α∑_k=1^m/4Γ_m-4kα(λ) mα+ρ-4kα,α-i2α, λ.

We can calculate

    ρ_α := ⟨ρ,α⟩/|α|^2 = 1/2 m_α  + m_2α;

moreover,

    m^2α^2-2miα, λ≥maxm^2α^2,2mα,λ.

This allows us to estimate Γ_mα(λ) as follows:
    
    Γ_mα(λ)   ≤ 2m_α∑_k=1^m/2Γ_m-2kα(λ)m+ρ_α-2k/m^2+1/2m
          + 2m_2α∑_k=1^m/4Γ_m-4kα(λ)m+ρ_α-4k/m^2+1/m
       ≤ 2m_αρ_α/m^2+3/2m∑_k=1^m/2Γ_m-2kα(λ)
    			+ 2m_2αρ_α/m^2+2/m∑_k=1^m/4Γ_m-4kα(λ).

With Γ_0≡1 and Γ_α≡0, this implies already that for every m there is a constant C_m≥1, independent of λ, such that |Γ_mα(λ)|≤ C_m. It is clear that every C_m depends on m_α and m_2α only.

Set C_α^0=max{ C_m: 0≤ m≤ρ_α} and C_α = max{ C_α^0, 1/25m_α + 3m_2α}. 
In particular, we have C_α≥5/2 since m_α≥1. 
We will show now by induction that |Γ_mα(λ)|≤ C_α^m for all m and all regular λ. 
The assumption is true for m≤ρ_α. 
For m>ρ_α, suppose the assumption holds for all l<m, then 

    Γ_m α(λ)   ≤ 2m_α ( 1/m + 3/2m) ∑_k=1^m/2Γ_m-2kα(λ)
    		+ 2m_2α ( 1/m + 2/m) ∑_k=1^m/4Γ_m-4kα(λ)
       ≤ 5m_α/m∑_k=1^m/2 C_α^m-2k
    		+  6m_2α/m∑_k=1^m/4 C_α^m-4k
       ≤ 5m_α/m m/2 C_α^m-2
    		+  6m_2α/mm/4  C_α^m-4
       ≤(  5m_α/2 +  3m_2α/2) C_α^m-1≤ C_α^m.
		


Then we can estimate spherical functions as follows:

    φ_λ(exp H)^ρ(H)   ≤(λ)∑_m=0^∞^-mα(H)Γ_mα(s_αλ)+Γ_mα(λ)
       ≤ 2 (λ)∑_m=0^∞^-mα(H) m^p,

the series converging for any H∈^+. 
Set 

    ^+_1=H∈^+: α(H)> 1.

We obtain that for any regular λ and H∈_1^+,

    φ_λ(exp H)^ρ(H)≤ C'(λ),

with

    C' = 2 ∑_m=0^∞ e^-m m^p.


For any H∈\_1^+=H∈: α(H)≤ 1, in virtue of φ_λ_∞=φ_λ(1)=1,

    φ_λ(exp H)^ρ(H)≤^ρ(H)· 1
    ≤^1/2m_α+m_2α =: C”
    < ∞.

Thus, for every H∈ and regular λ, we have |φ_λ(exp H)|^ρ(H)≤ C 

max(1,|(λ)|),
with C=max(C', C”).

The set ^*∖^*' of irregular points has measure zero in ^* and does not influence integration. 





Together with (<ref>),	
we now obtain

    sup_a∈∫_^*R(λ)φ_λ(a)^ρ(log a)(λ)^-1d λ≤ C ∫_^*R(λ)1+λ^n-l/2dλ,


addressing both assertions of the theorem. 

	
Step 2.
We induct on Σ_s^+. 

If Σ_s^+=1, by the discussion in Step 1, the result follows.
Assume now the theorem holds 
for all Lie groups of class ℋ whose simple system satisfies Σ_s^+≤ r-1.





We proced to consider the case Σ_s^+= r.

We denote by Z_ the centre of  (which may be nontrivial), and set =∩ Z_.	 
As a subspace of ,

    =⋂_α∈Σ^+α=⋂_α∈Σ_s^+ α.

For each positive root α we can define the quotient linear functional α̅: /→ by

    α̅(H̅):=α(H), H̅ = H+∈/ .

It is clear that p(H̅):=max_α∈Σ^+_s α̅(H̅) is a norm on /.

Moreover, it is equivalent to the quotient norm since / is finite-dimensional.
It follows that
    
 
	
 
we can find a constant ζ>0 such that p(H̅)>ζH̅ for any H∉.
Recall that the simple system is Σ_s^+=α_1,⋯,α_r. 
For each H∉ there exists α_k∈Σ_s^+ such that p(H̅)=α̅_k(H̅) > ζH̅. 
By definition of the quotient norm, this implies that there exists Y∈ such that α_k(H)=α_k(H+Y)>ζH+Y.
Moreover, if H∈ then α_k(H)=α_k(H+Y)> 0 and also H+Y ∈.
Define _k^+=X∈: α_k(X)> ζX, then H=H+Y-Y∈_k^+ + and we have just shown that 

    ⊂⋃_k=1^r (^+_k+)∪.


We choose next a basis H̅_1,…, H̅_r of / dual to α̅_1, …, α̅_r and pick representatives H_k∈ of H̅_k such that α_k(H_k) > ζH_k. This ensures in particular that α_j(H_k)=0 if j k, and H_k∈_k^+⊂.




Step 3.

For any H∈, α∈Σ^+ and Y∈_α, we get α(H)Y=H,Y=0. 
It implies ρ(H)=0 and δ(a)=^-2ρ(H)=1.
With the fact φ_λ_∞ = 1 we have

    sup_a∈exp∫_^*R(λ)φ_λ(a)^ρ(log a)(λ)^-1d λ≤∫_^*R(λ)1+λ^n-l/2 dλ .


Next, estimates on ^+_k+ are the same as on ^+_k, which is verified as follows.
By <cit.> we have for any Y∈ and H∈, 

    φ_λ(exp H)
    		   =φ_λ(exp(-Y+Y+H))
       =∫_K ^(-iλ+ρ)(A(kexp Y))^(iλ+ρ)(A(kexp(Y+H)))dk
       =∫_K ^(-iλ+ρ)(A(exp Yk))^(iλ+ρ)(A(kexp(Y+H)))dk
       =∫_K ^(-iλ+ρ)(Y)^(iλ+ρ)(A(kexp(Y+H)))dk
       =^-iλ(Y)∫_K ^(iλ+ρ)(A(kexp(Y+H)))dk
        = ^-iλ(Y)φ_λ(exp(Y+H)).

Thus we obtain 

    φ_λ(exp(H+Y))^ρ(H+Y)
        =φ_λ(exp H)^ρ(H),

that is,

    sup_H∈^+_k+ |φ_λ(exp H)|^ρ(H)
     = sup_H∈^+_k |φ_λ(exp H)|^ρ(H).



Step 4.
For each k∈1,…,r, set F_k = {α_j: j k}. This is exactly the set of simple roots of G vanishing on H_k. Let P_k=P_F_k be the parabolic subgroup associated with F_k, as described in Section <ref>. Denote by G_k=M_1F_k the associated subgroup of P_k. The simple system on the Lie algebra _k of G_k is exactly F_k and trivially F_k=r-1. We write for convenience Σ_k^+:=Σ_F_k^+, Σ_k:= Σ_F_k, _k:=_F_k, _k:=_F_k, τ_k:=τ_F_k and ρ_k:= ρ_F_k= 1/2∑_α∈Σ_k^+ m_αα. Then the conical region (<ref>) corresponding to H_k is

    A^+(H_k:ζ)=a∈: τ_k(log a)> ζ |log a|=a∈: α_k(log a) > ζ |log a|.

The sets (_k^+) can be represented now in the following form: 

    _k^+ =log x: x∈ A^+(H_k: ζ).

    
According to Notation <ref> and to Theorem <ref>, for every k there exist constants C_k(ζ), ι_k, κ_k such that for any a∈ A^+(H_k: ζ)

    φ_λ(a)^ρ(log a)   ≤^ρ_k(log a)ψ_λ(a)+C_k(ζ) 1+λ^ι_k^-κ_k log a
       = _k^-1∑_s∈(sλ)/_k(sλ)θ_sλ(a)^ρ_k(log a)+C_k(ζ)1+λ^ι_k^-κ_k log a;

by Corollary <ref>, we can assume ι_k = 6d+1. 
Note also that ζ is the same for all k.
We denote then

    I_k'(a)    = ∫_^*R(λ)_k^-1∑_s∈(sλ)/_k(sλ)θ_sλ(a)^ρ_k(log a)(λ)^-1d λ,
      
    
      I_k”(a)    = C_k(ζ) ∫_^*R(λ)1+λ^ι_k^-κ_k |log a|(λ)^-1d λ

and

    I_k(a) = I_k(a)'+ I_k(a)”.


    :=sup_a∈ A^+(H_k: ζ)∫_^*R(λ)φ_λ(a)^ρ(log a)(λ)^-1d λ
       ≤∫_^*R(λ)_k^-1∑_s∈(sλ)/_k(sλ)θ_sλ(a)^ρ_k(log a)(λ)^-1d λ
            + C_k(ζ)∫_^*R(λ)1+λ^ι_k^-κ_k |log a|(λ)^-1d λ
       =:I_k'+ I_k”.



To estimate the main part I_k(a)', recall first that |(sλ)|=|(λ)| for each s∈, so that

    I_k'(a)
    		   ≤∫_^*R(λ)_k^-1∑_s∈(sλ)/_k(sλ)θ_sλ(a)^ρ_k(log a)(λ)^-1 dλ
       =  ∫_^*R(λ)_k^-1∑_s∈_k(sλ)^-1θ_sλ(a)^ρ_k(log a) dλ.

Next, every s∈ is an orthogonal transformation of , which implies


R(λ)=R(λ)=R(sλ)=R(sλ). 
Changing the variables, we can now estimate the integral I_k'(a) as follows:

    I_k'(a)
    		   ≤ |_k|^-1∑_s∈∫_^*R(sλ)  _k(sλ)^-1θ_sλ(a)^ρ_k(log a) dλ
       = 1/|_k|∑_s∈^ρ_k(log a)∫_^*R(λ) _k(λ)^-1θ_λ(a) dλ
       = ||/|_k| ^ρ_k(log a)∫_^*R(λ)θ_λ(a)_k(λ)^-1dλ.

The simple roots of G_k with respect to  are {α_j: j k} and there are r-1 of them. The number d_k of indivisible roots is smaller than d. We can apply thus the inductive hypothesis to get
    
    sup_a∈ I_k'(a)
                
                ≤ C_k∫_^*R(λ)1+λ^6d_k+1+n_k-l/2 dλ

and

    lim sup_|log a|→∞  I_k'(a) 
    ≤ C_k∫_^*R(λ)1+λ^n_k-l/2 dλ,

with a constant C_k depending on the group G_k only.

The remainder term I_k”(a) is bounded by

    I_k”(a) ≤ C_k(ζ)c_G_k∫_^*R(λ)(1+|λ|)^6d+1+(n-l)/2 d λ ^-κ_k |log a|
    ;

it contributes to the uniform norm but not to the supper limit at |log a|→∞.

    Thus, keeping the largest powers only, 
    
    sup_a∈ A^+(H_k: ζ) I_k(a)
    
    
    
    
                   ≤ C_k' ∫_^*R(λ)1+|λ|^6d+n-l/2+1 dλ


and

    lim sup_a∈ A^+(H_k: ζ)  |log a|→∞  I_k(a) 
    ≤ C_k'∫_^*R(λ)1+λ^n-l/2 dλ

with a constant C_k'≥1.
For any a∈, let X=log a, if X∈, the inequality (<ref>) is obtained in the Step 3. If X∉, by the cover of positive chamber (<ref>), there exists k∈1,…, r such that X∈_k^+ +. Let us write X=H+Y where H∈_k^+ and Y∈. With the relation (<ref>), we obtain
    
    sup_X∈_k^+ +∫_^*R(λ)φ_λ(exp X)^ρ(X)(λ)^-1d λ
        = sup_H∈_k^+∫_^*R(λ)φ_λ(exp H)^ρ(H)(λ)^-1d λ
       ≤ C∫_^*R(λ)1+λ^z_k + n-l/2 dλ



With (<ref>), it remains now to collect (<ref>), (<ref>) and the inequality above to get 
the statement of the theorem, with C=max{ C'_k: 1≤ k≤ r}.










For every H∈ we have one of the following possibilities:


  * H is regular, and all simple roots are nonzero on it. Then on every induction step it falls into one of the regions ^+(H_k:ζ) (to think again if ζ changes from step to step or can be estimated once for the whole of the proof). This continues until there is only one simple root left; and in this case we are in Step 1. The only case is possible:

    
  * H∈^+_1,α, then the bound for this step is C|(λ)|;



  * The second case is when α_j(H)=0 for j∈ F⊂Δ^+_s which is nonempty. Then at certain point of induction we fall into Step 3 and get 1 as the bound.

The remainders are of order (1+|λ|)^ι e^-κ |H| in both cases.
That is, we get no decay in H, what is somewhat surprizing. Probably this is because we do not use any differentiation conditions.


From this moment we suppose that G is a semisimple connected Lie group with finite center and F a Borel function on ^+.
We proceed 
to obtain uniform estimates for the kernel of F(L), and by consequence pointwise bounds for the kernel of F(Δ_ρ). The following lemma shows that it is sufficient to estimate the values on the subset ⊂ G only.




Let k_ be defined as in Section <ref>, then
    
    k__∞=sup_x∈ Gk_(x)≤sup_a∈k_(a).



	Every x∈ G can be decomposed as x=k_1 a k_2 where a∈ and k_1, k_2∈ K. 
	In virtue of (<ref>) and knowing that k_ is K-biinvariant, we obtain
	
    k_(x)
    			   =δ^-1/2(k_1 ak_2)k_(k_1ak_2)
       =δ^-1/2(ak_2)k_(a)
       =δ^-1/2(ak_2)δ^1/2(a)δ^-1/2(a)k_(a)
       = ^ρ(H(ak_2)-log a)k_(a).

	By <cit.> (see also <cit.>) we know that
	
    ρ(H(ak))≤ρ(log a)  for all a∈, k∈ K.

	Then we get 
	
    ^ρ(H(ak_2)-log a)≤ 1
 
	and 
	
    sup_x∈ Gk_(x)≤sup_a∈k_(a).

	This completes the proof.




Recall that x^+ denotes the  component in the polar decomposition of x∈ G.

    Let G be a semisimple connected Lie group with finite center and F a Borel function on ^+ such that 
    
    I_F =    ∫_^*F(λ^2) (1+|λ|)^6d+n-l+1 dλ < ∞.

    Then k_ is uniformly bounded on G, and 
    
    k__∞ 
        =sup_x∈ Gδ^-1/2(x)∫_^* F(λ^2)  (λ)^-2φ_λ(x) dλ≤ C I_F,

    where C is a constant depending on G only.
    Moreover,
    
    lim sup_|log x^+|→∞ |k_(x)| ≤ C ∫_^* |F(|λ|^2)| (1+|λ|)^n-l dλ.



    By Lemma <ref>, it is sufficient to estimate |k_(a)|





for a∈.
    Since G is semisimple it is also in class ℋ; we take 
    
    R(λ)=F(λ^2)1+λ^n-l/2

    in Lemma <ref> to obtain, using (<ref>), that
	
    sup_a∈k_(a)   ≤sup_a∈∫_^*F(λ^2)^ρ(log a)φ_λ(a)(λ)^-2 dλ
    
    		   ≤sup_a∈∫_^* R(λ)  ^ρ(log a)φ_λ(a)(λ)^-1 dλ
       ≤ C∫_^* R(λ) (1+|λ|)^6d+1+n-l/2d λ
       = C∫_^* |F(|λ|^2)| 1+|λ|^6d+n-l+1d λ.

	The second inequality is obtained similarly.



In the statement of the theorem, one can clearly replace (<ref>) by    
    
    I_F =    ∫_^*F(λ^2) (1+|λ|)^7(n-l)+1 dλ < ∞.




    Let G be a semisimple connected Lie group with finite center and F a Borel function on ^+ verifying (<ref>) or (<ref>).
    Then  for every x∈ G
    
    |k_(x)|  ≤ CI_F  e^ -ρ( H(x) ) ,

    where C is a constant depending on G only.
    Moreover,
    
    lim sup_|log x^+|→∞ |k_(x)| e^ρ( H(x) ) ≤ C ∫_^* |F(|λ|^2)| (1+|λ|)^n-l dλ.
    




It follows from Theorem <ref> that the operator F(L) is bounded from L^1(G,m_r), with respect to the right Haar measure, to L^∞(G). This occurs frequently, as soon as the integral (<ref>) converges,  without any smoothness or positivity assumptions on the symbol.









One can also note that from L^1(G,m_l) to L^∞(G) these operators are never bounded, unless F is zero (this is a general fact about left convolution operators). 

This can be seen as follows: [to be shortened] For any x we have

    (k*f)(x)    = ∫ k(y) f(y^-1x) dx
    = ∫ k(xz) f(z^-1) dz
    = ∫ k(xz^-1) f(z) δ(z)^-1 dz
    
        = ∫ k(xz^-1) δ(xz^-1) δ(x^-1) f(z) dz.

The norm of the linear functional v_x: f↦ (k*f)(x) is

    esssupv_x = _z k(xz^-1) δ(z^-1) = δ(x^-1) _z k(xz^-1) δ(xz^-1) 
    = δ(x^-1) _y k(y) δ(y);

as soon as k≢0, we have kδ_∞>0 and

    F(L)_L^1→ L^∞ = sup_x,f |(k*f)(x)| /f_1  = sup_x v_x = +∞.




  
From the known estimate <cit.> 
of spherical functions

    |φ_λ(e^H)| ≤ C (1+H)^d e^-ρ(H), for all H∈


it is easy to obtain estimates for the kernel of the type |k_F(L)(e^H)| ≲ (1+H)^d and |k_F(Δ_ρ)(e^H)| ≲ (1+H)^d e^-ρ(H). Our result is that it is possible to remove this polynomial factor.
In the analysis of the Laplace-Beltrami operator Δ, a factor of this kind is often non-significant. But it becomes essential in concern with the operator L.






§ LOWER BOUNDS FOR OSCILLATING FUNCTIONS IN RANK ONE




  §.§.§ Higher ranks


Fix H_k∈ such that α_k(H_k)=1 and α_j(H_k)=0 if j k. In the notations of Section 2 (?? link), we have ρ_0(H_k)=0, and moreover, H_k is in the center of the Lie algebra 𝔪_10. It follows that θ_λ(exp (u H_k)) = e^iuλ(H_k) for every u∈. By Theorem 2.7 (link?),

    |φ_λ(e^uH_k) - _0^-1∑_s∈(sλ)/_0(sλ) e^iusλ(H_k)|
    < C(1+λ)^ι^-κ |uH_k|

with some constants ι, C, κ.
Simple roots of _10 are α_j, j k, so that _0 is the subgroup of  generated by s_α, where α is in the linear span of α_j, j k.

[We might have roots of the type ∑ l_j α_j with l_k0 but also l_j0 for some other j, this will influence the exponential but also the c-function.]



  §.§.§ Rank one



Estimates of Section 4 involve the absolute value of the function F and may seem unsatisfactory for oscillating functions of the type F(x) = exp(it√(x))  ψ(√(x)), where one would hope to see a decrease in t as t→∞. However, they are optimal even in this case, if G has rank one. We show this in the present section. In higher rank the behaviour would be different, but it will be considered elsewhere.

Let us assume now that G is a semisimple Lie group of real rank one.

Both  and ^* can be viewed then as the real line  and Σ_s^+=1. Let α be the only positive root; we can assume that it acts at H∈≃ as simple multiplication, α(H) = α H.

For all H∈^+ and all regular λ∈, we have the Harish-Chandra decomposition (<ref>) of φ_λ(exp H)^ρ(H).





By <cit.>, there exist constants C_α>0, p>0 such that |Γ_mα(λ)| ≤ C_α m^p for all λ∈^*. For H such that α(H)>1, we can then estimate


    φ_λ(exp H)^ρ(H)-  (λ)^iλ(H)- (s_αλ)^is_αλ(H)
         ≤ C_α( |(λ)| + |(s_αλ)| ) ∑_m=1^∞ m^p ^-mα(H)
            < C |(λ)| e^-α(H).

Since s_αλ= -λ for all λ∈^*, we have

    k_(exp H)   = ∫_ F(λ^2) φ_λ(exp H)^ρ(H) (λ)^-2d λ
       ≥∫_ F(λ^2) (λ)^-2 (λ)^iλ(H) +  (-λ)^-iλ(H) d λ
          - C ∫_F(λ^2)^-α(H) | (λ)|^-1 d λ.


Now we set F(x)=^it√(x)ψ(√(x)) where t>0 and ψ is a continuous function on  such that 

    J_ψ = ∫_0^∞ |ψ(x)| (1+x)^n+6 dx < ∞.

This corresponds to (<ref>) in rank one when d=l=1.
Denote by k_t the kernel of the operator F(L), then under the assumptions above on H we have k_t(exp H)≥I_1(t,H)-I_2(H) where

    I_1(t, H)= ∫_^itλψ(λ)  (λ)^-2 (λ)^iλ(H) +  (-λ)^-iλ(H) d λ,


    I_2(H) = C ^-α(H)∫_ψ(λ) | (λ)|^-1 d λ.

For the asymptotics of k_t with large H, we only need to consider the I_1(t,H), as I_2(H) vanishes as H→ +∞ under the condition (<ref>). Let us write H=t+a with a∈. Using the conjugation property (<ref>) of the -function, we transform I_1 as follows:


    1/2  I_1(t,t+a)
               =∫_^itλψ(λ) (λ)^-2 (λ)^iλ (t+a) d λ
       =∫_0^∞^itxψ(x)(x)^-2(x)^i(t+a)x+(-x)^-i(t+a)x d x
       = 2∫_0^∞ψ(x)(x)^-2^itx(x)^i(t+a)x d x.

After elementary calculations this takes form

    = ∫_0^∞ψ(x)(x)^-2[ (x) ( (1+cos(2tx)) cos(ax) - sin(2tx) sin(ax)
          + isin(2tx) cos(ax) -i (1-cos(2tx)) sin(ax) )
          - (x) ( sin(2tx) cos(ax) + (1+cos(2tx)) sin(ax) 
          + i (1-cos(2tx)) cos(ax) + i sin(2tx) sin(ax) ) ] d x,

which tends at t→+∞ to

    ∫_0^∞ψ(x)(x)^-2[ ( (x) cos(ax) - (x) sin(ax) )
          -i ( (x) sin(ax) + (x) cos(ax) ) ] d x
       = ∫_0^∞ψ(x)(x)^-2[ ( (x) e^iax) - i( (x) e^iax) ] d x
       =∫_0^∞ψ(x)(x)^-2(x)  e^-iax d x
       =∫_0^∞ψ(x)(x)^-1 e^-iax d x .

This is the Fourier transform at a of the function ψ̃ which is equal to ψ(x)(x)^-1 if x≥0 and 0 otherwise. Note that ψ̃∈ L^1 since ψ satisfies (<ref>).
For ψ≢0, this Fourier transform (ψ̃) is a nonzero continuous function vanishing at infinity, so its norm ν=(ψ̃)_∞>0 is attained at a point a_0∈. 
For t≥α^-1-a_0, we have α(H)=α(a_0+t)≥ 1, and the estimate above is applicable. Altogether, this implies that 

    lim inf_t→ +∞k_t_∞≥lim_t→ +∞k_t(exp (t+a_0))≥lim_t→ +∞(I_1(t,t+a_0)-I_2(t+a_0))=ν >0.

In particular, this applies to the most common localization function ψ(x)=1+x^2^-κ where κ is a constant such that (<ref>) holds. 
To summarize the above, we proved the following theorem:

    Let G be a semisimple Lie group of real rank one. Denote by k_t the kernel of ^it√(L)ψ(√(L)) where t>0 and ψ is a continuous function on  satisfying (<ref>).
    
    
    
    
    
    
    Then k_t_∞≤ CJ_ψ for all t, and lim_t→ +∞k_t_∞≥ν where ν=(ψ̃)_∞>0 and
    
    ψ̃(x) = 
        ψ(x)(x)^-1   if x≥ 0,
    
        0    if x< 0.

    In particular, if ψ(x)=1+x^2^-κ, then the theorem applies if κ>(n+7)/2.




§ UNIFORM NORMS OF OSCILLATING FUNCTIONS


 

  * In rank one: Tataru <cit.>: the symbol is sin(λ t)λ (λ^2+β^2)^-ρ/2+is so that the operator is 
sin(√(|Δ_ρ|) t)√(|Δ_ρ|) (Δ_ρ+β^2)^-ρ/2+is, on the hyperbolic space (ℓ=1), the uniform bound is c(sinh t)^-ρ.

  * Anker, Pierfelice (on Damek-Ricci): D^-τD̃^τ-σ e^itD where D=√(|Δ|-ρ^2), D̃=√(|Δ|+ρ̃^2-ρ^2), ρ̃>ρ, σ∈_+, τ∈[0,3/2). That is, the operator is
 
    (|Δ|_ρ)^-τ/2 (|Δ|_ρ+ρ̃^2)^(τ-σ)/2 e^it√(|Δ|_ρ)

 <cit.>
 
  * Higher rank: Hassani <cit.>: |Δ|^-σ/2 e^it√(|Δ|); the bounds are for |t|≥1:

 |k_0(x)| ≤ C |t|^-l/2φ_0(x) (3+|x|)^a if a∈ 2 and l<a (integral over |λ|≤1);

 |k_∞(x)| ≤ C |t|^-dϕ_0(x) (3+|x|)^d if σ >n (integral over |λ|≥1).
 
  * Anker and Zhang <cit.>: |Δ|^-σ/2 e^it√(|Δ|), d=ℓ+∑_α∈Σ^+ m_α≥3 (but ℓ may be 1). Boundary point: σ = (d+1)/2. Since d=2n-l, this value is n-l/2+1/2.

 Theorem 3.7: for large t and |x|≥|t|/2, the bound is |t|^-N_1 (1+|x^+|)^N_2 e^-ρ(x^+) for every N_1∈ and N_2>N_1+2(d+1)+ (1/2) (max(d,D) -l).
 
 Theorem 3.3: For |x/t|<C_Σ<1/2, |ω̃^σ,0_t(x)| ≤ |t|^-D/2 (1+|x|)^(D-l)/2ϕ_0(x).
 

The maximum on [0,+∞) of the function f(x) = x^a e^-bx (with a,b>0) is ( abe)^a.
[details to remove: f'(x) = x^a-1 e^-bx ( a -bx) and vanishes at x=a/b, this is clearly the maximum and f(a/b) = ( ab)^a e^-a.] In the expression t^-N_1 x^N_2 e^-ρ x) (assuming x≥ t>0) set x=t^γ, supposing γ≥1; then we get

    t^γ N_2-N_1 e^-ργ t

whose maximum on (0,+∞) is attained at ργγ N_2-N_1 and equal to

    ( abe)^a





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