On a restriction problem of de Leeuw type for Laguerre multipliers

Abstract

In 1965 K. de Leeuw \cite{deleeuw} proved among other things in the Fourier transform setting: {\it If a continuous function $m(ξ_1, \ldots ,ξ_n)$ on ${\bf R}^n$ generates a bounded transformation on $L^p({\bf R}^n),\; 1\le p \le \infty ,$ then its trace $\tilde{m}(ξ_1, \ldots ,ξ_m)=m(ξ_1, \ldots ,ξ_m,0,\ldots ,0), \; m<n,$ generates a bounded transformation on $L^p({\bf R}^m)$. } In this paper, the analogous problem is discussed in the setting of Laguerre expansions of different orders.