A Sobolev estimate for radial $L^p$-multipliers on a class of semi-simple Lie groups

Abstract

Let $G$ be a semi-simple Lie group in the Harish-Chandra class with maximal compact subgroup $K$. Let ${\Omega }_K$ be minus the radial Casimir operator. Let $\frac{1}{4}\dim (G/K)<S_G<\frac{1}{2}\dim (G/K),s\in (0,S_G]$ and $p\in (1,\infty )$ be such that \[ \left| \frac{1}{p} - \frac{1}{2} \right| < \frac{s}{2 S_G}. \] Then, there exists a constant $C_{G,s,p}>0$ such that for every $m\in L^{\infty }(G)\cap L^2(G)$ bi-$K$-invariant with $m\in \mathrm{Dom}({\Omega }_K^s)$ and ${\Omega }_K^s(m)\in L^{2S_G/s}(G)$ we have, \[ \Vert T_m: L^p(\widehat{G}) \rightarrow L^p(\widehat{G}) \Vert \leq C_{G, s,p} \Vert Ω_K^s(m) \Vert_{L^{2S_G/s}(G)}, \] where $T_m$ is the Fourier multiplier with symbol $m$ acting on the non-commutative $L^p$-space of the group von Neumann algebra of $G$. This gives new examples of $L^p$-Fourier multipliers with decay rates becoming slower when $p$ approximates $2$.

arXiv commentTo appear in Transactions of the AMS