Fourier multipliers and group von Neumann algebras

Abstract

In this paper we establish the $L^p$-$L^q$ boundedness of Fourier multipliers on locally compact separable unimodular groups for the range of indices $1<p\le 2\le q<\infty$. Our approach is based on the operator algebras techniques. The result depends on a version of the Hausdorff-Young-Paley inequality that we establish on general locally compact separable unimodular groups. In particular, the obtained result implies the corresponding Hörmander's Fourier multiplier theorem on ${\mathbb{R}}^n$ and the corresponding known results for Fourier multipliers on compact Lie groups.

Comptes Rendus Math. (to appear)

References in corpus (4)

• Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and Lp-Lq Fourier multipliers on compact homogeneous manifolds
• $L^p$-$L^q$ multipliers on locally compact groups
• Hardy-Littlewood inequalities and Fourier multipliers on SU(2)
• Pseudo-differential operators, Wigner transform and Weyl systems on type I locally compact groups

Cited by in corpus (3)

• Smooth dense subalgebras and Fourier multipliers on compact quantum groups
• $L^p$-$L^q$ multipliers on locally compact groups
• The weak type $(1,p)$ for convolution operators on locally compact groups