Lp multipliers on non compact symmetric spaces

Let G be a noncompact connected semisimple Lie group, with finite center, K a maximal compact subgroup of G, g=k⊕p be a Cartan decomposition of the Lie algebra of G, a a maximal abelian subspace of p. We shall denote by n the dimension of the homogeneous space X=G/K and by r its rank. It is well known that a spherical multiplier of Lp(X), for some p, 1<p<∞, extends to a bounded, Weyl invariant holomorphic function in an open tube T_{|2/p - 1|} in the complex dual of a. In this paper we investigate in some particular cases (G complex, r=1) relationship between classical Fourier multipliers of Lp on the edge of the tube and spherical multipliers of Lp(X). The gist of our result is the following: there exists a function ω, holomorphic and non vanishing in a neighbourhood of the tube T_1, which grows of order (n-r)/2 at infinity, such that, if the restriction of ωm to the edge of T_1 is a Fourier multiplier of Lp of the edge, for all p, 1<p<∞, then m is a spherical multiplier of Lp(X). By a standard argument of complex interpolation if m satisfies a similar condition on the edge of the tube T_{|2/q - 1|}, 1<q<2, then m is a spherical multiplier of Lp(X) for all p, q<p<q'. Thus by imposing a decay condition on the edge of the tube, we are able to consider a larger class of multipliers whose boundary values are less regular.