Hausdorff-Young-Paley inequalities and $L^p$-$L^q$ Fourier multipliers on locally compact groups
In this paper we establish the $L^p$-$L^q$ boundedness of Fourier multipliers on locally compact separable unimodular groups for the range for $1<p\leq 2 \leq q<\infty$. For this, we also prove a version of the Hausdorff-Young-Paley inequality on general locally compact separable unimodular groups. In particular, the obtained result implies H\"ormander's Fourier multiplier theorem on $\mathbb{R}^{n}$ and the corresponding known results for Fourier multipliers on compact Lie groups.