M.El Hamma,R.Daher and A.Akhlidj

LaboratoireTAGMD,FacultédesSciencesA?nChock,UniversitéHassanII,B.P5366 Maarif,Casablanca,Maroc

Abstract.Two estimates useful in applications are proved for the Fourier transform in the space L2(X),where X a symmetric space,as applied to some classes of functions characterized by a generalized modulus of continuity.

Key Words:Fourier transform,generalized continuity modulus,symmetric space.

1 Introduction and preliminaries

In[2],Abilov et al.proved two useful estimates for the Fourier transform classic in the space of square integrable functions on certain classes of functions characterized by the generalized continuity modulus,using a translation operator.

In this paper,we prove the analog of Abilov’s results see[2]in the Fourier transform on rank 1 symmetric space.

Let X=G/K where G is a connected noncompact semisimple Lie group with finite center and real rank one and K is a maximal compact subgroup.The form of Cartan decomposition is defined by g=e+p,where e is the Lie algebra of K.And g=e+a+n is Iwasawadecomposition,where a isamaximalabeliansubalgebraof p and n isanilpotent subalgebra of g.The rank one condition is that dima=1.the nilpotent subalgebra n has root space decomposition n=nγ+n2γ,where γ and 2γ are the positive roots.Let mγand m2γbe the respective root space dimensions and set(mγ+2m2γ).Choose H0∈a such that γ(H0)=1.This allows identifying a with R by the map t∈R←?tH0∈a,and denote a?the real dual space of a.

Let G=NAK be the Iwasawa decomposition of the group G,and g,e,a,and n the respective Lie algebras of the groups G,K,A,and N.Denote by M the centralizer of the subgroup A in K and put B=k/M.Let dx be a G-invariant measure on X,and let db and dk be the respective normed K-invariant measure on B and K.

The finiteWeylgroupWactson a?.Supposethat∑ isthesetofboundedroots(∑?a?),∑+is the set of positive bounded roots,and a+={h∈a;α(h)>0 for α∈∑+}is the positive Weyl chamber.Leth,ibe the Killing form on the Lie algebra g.For λ ∈a?we denote by Hλthe vector in a such that λ(H)=hHλ,Hifor all H in a.Let

The dimension of X is equal to

We return to the case in which X=G/K is an arbitrary symmetric space.Given g∈G,denote by A(g)∈a the unique element satisfying

where u∈K and n1∈N.For x=gK∈X and b=kM∈B=K/M,we put

In terms of this decomposition,the invariant measure dx on X has the form

where ?(t)= ?(α,β)(t)=(2sinht)2α+1(2cosht)2β+1,α =(mγ+m2γ?1)/2 and β =(m2γ?1)/2.The Laplacian on X is denoted L and its radial part is given by

The spherical function on X is the unique radial solution to the equation

The spherical function is defined by

where jα(t)is a normalized Bessel function of the first kind.

Proof.See[3],Lemma 9.

Lemma 1.2.The following inequalities are valid for a Jacobi function φλ(t)(λ,t∈R+):

1.|φλ(t)|≤1.

2.|1?φλ(t)|≤t2(λ2+ρ2).

Proof.See[6],Lemmas 3.1-3.2.

Consider in the Hilbert space L2(X)=L2(X,dx)with the norm

The Fourier transform on X was introduced by S.Helgason(see[4,5]),and defined for all f∈L2(X)by the formula

The inverse the Fourier transform

where|W|is the order of the Weyl group,and dμ(λ)=|c(λ)|?2dλ with dλ is the element of the Euclidean measure on a?,and c(λ)is the Harish-Chandra function.

The Fourier transform is an isomorphism of the Hilbert space L2(X)onto the Hilbert space L2(×B,dμ(λ)db).

The Palancherel formula

Introduce the translation operator on X.Denote by d(x,y)the distance between x,y∈X and let

be the sphere of radius h>0 on X centered at x.Let dσx(y)be the(dimX?1)-dimensional area element of the sphere σ(x,h)and let|σ(h)|be the area of the whole sphere σ(x,h).

Let f∈L2(X),the translation operator Shis defined by

From[7],we have

Lemma 1.3.Let f(x)∈L2(X),then

Proof.See[6],Lemma 1.4.

The finite differences of the first and higher orders are defined as follows:

is the kth-order generalized continuity modulus of f∈L2(X).

and ψ(t)is an arbitrary function defined on[0,∞).

From formula(1.1),we have

and

By Parseval identity,we obtain

To make the formulas concise,we introduce the notation

Using the Lemma 1.3 and it is easy to show tha for f∈L2(X)

2 Estimates for the Fourier transform

Theorem 2.1.For functions f(x)∈L2(X)in the class

where m=0,1,···,k=1,2,···,;c>0 is a fixed constant,and ψ(t)is any function defined on the interval[0,∞).

Proof.In the terms of jp(x)a normalized Bessel function of the first kind we have(see[1])

where Jp(x)is Bessel function of the first kind,which is related to jp(x)by the formula

From the formula(1.2),we have

Then,

Combining this with formulas(2.1c)and(2.2),we have

Or

Then

This completes the proof of theorem.

Theorem 2.2.Let ψ(t)=tν,where(ν>0),then

m=0,1,2,···,k=1,2,···,0<ν<2.

Proof.We prove sufficiency by Theorem 2.1:letand ψ(t)=tνthen

To prove necessity let

It is easy to show,that there exists a function f∈L2(X)such that Lf∈L2(X)and

Hence,by the Plancherel equality,we have

This integral is divided into two

where N=[h?1].We estimate them separately

i.e.,

Now,we estimate I1,by(2)in Lemma 1.2,we obtain

i.e.,

Combining the estimates for I1and I2gives

Acknowledgements

The authors would like to thank the referee for his valuable comments and suggestions.