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

Anker, Jean-Philippe

Ann. Math. (2) 132, No. 3, 597-628 (1990).

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.

Keywords:

Riemannian symmetric space; bounded operators; biinvariant distribution; Fourier transform; multiplier; spherical Fourier transform; multiplier theorem; Abel transform