$L^p$-$L^q$ multipliers on locally compact groups

In this paper we discuss the $L^p$-$L^q$ boundedness of both spectral and Fourier multipliers on general locally compact separable unimodular groups $G$ for the range $1<p\leq q<\infty$. We prove a Lizorkin type multiplier theorem for $1<p\leq q<\infty$, and then refine it as a H\"ormander type multiplier theorem for $1<p\leq 2\leq q<\infty$. In the process, we establish versions of Paley and Hausdorff-Young-Paley inequalities on general locally compact separable unimodular groups. As a consequence of the H\"ormander type multiplier theorem we derive a spectral multiplier theorem on general locally compact separable unimodular groups. We then apply it to obtain embedding theorems as well as time-asymptotics for the $L^p$-$L^q$ norms of the heat kernels for general positive unbounded invariant operators on $G$. We illustrate the obtained results for sub-Laplacians on compact Lie groups and on the Heisenberg group. We show that our results imply the known results for $L^p$-$L^q$ multipliers such as H\"ormander's Fourier multiplier theorem on $\mathbb{R}^{n}$ or known results for Fourier multipliers on compact Lie groups. The new approach developed in this paper relies on the analysis in the group von Neumann algebra for the derivation of the desired multiplier theorems.