## $$\mathbb{L}_ p$$ Fourier multipliers on Riemannian symmetric spaces of the noncompact type.(English)Zbl 0741.43009

Let $$G/K$$ be a Riemannian symmetric space of non-compact type. One is interested in the bounded operators $$T$$ on $$L^ p(G/K)$$ which are $$G$$- invariant. Such an operator can be written $$Tf=f*\kappa$$, where $$\kappa$$ is a $$K$$- biinvariant distribution on $$G$$. By using the Fourier transform one obtains $$\widehat{Tf}(\lambda)=m(\lambda)\hat f(\lambda)$$, where the multiplier $$m(\lambda)$$ is the spherical Fourier transform of $$\kappa$$. The multiplier problem consists in describing the space $${\mathcal M}_ p$$ of multipliers of $$L^ p(G/K)$$. In this paper the author solves the problem by proving the following theorem, which is similar to the Hörmander-Michlin multiplier theorem for $$\mathbb{R}^ n$$.
Let $$1< p < \infty$$, $$v=|{1\over p}-{1\over 2}|$$, $$N=[v\dim G/K]+1$$. The function $$m$$ belongs to $${\mathcal M}_ p$$ if and only if (a) $$m$$ extends to a holomorphic function inside the tube $$T^ v={\mathfrak a}^* + i \hbox{conv}(2vW\rho)$$, (b) $$\nabla^ im$$ ($$i=0,\dots,N$$) extends continuously to the whole of $$T^*$$, with $$\hbox{sup}(| \lambda |+1)^{-i}|\nabla^ im(\lambda)| < \infty$$.
(The notations are the standard ones.) The multiplier problem has been solved previously in special cases: by J. L. Clerc and E. M. Stein if $$G$$ is a complex group, by R. J. Stanton and P. A. Tomas in the rank one case, and by M. E. Taylor if $$m$$ is a radial function. The most original idea in the proof is to use a support theorem for the Abel transform for the following polyhedral balls in $${\mathfrak a}$$, $${\mathcal V}_ p=\{H\in{\mathfrak a}\mid (wp)(H)\leq |\rho| r, w\in W\}$$, and a decomposition of $${\mathfrak a}$$ into shells $${\mathcal V}_{j+1}\backslash{\mathcal V}_ j$$.
Reviewer: J.Faraut (Paris)

### MSC:

 43A85 Harmonic analysis on homogeneous spaces 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc. 53C35 Differential geometry of symmetric spaces 43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
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