A Hausdorff-Young inequality for measured groupoids

TL;DR

This paper extends the classical Hausdorff-Young inequality from locally compact groups to measured groupoids using non-commutative $L^p$-spaces, broadening its applicability in harmonic analysis.

Contribution

It introduces a new Hausdorff-Young inequality for measured groupoids by leveraging the theory of non-commutative $L^p$-spaces and the structure of measured groupoids.

Findings

Extension of Hausdorff-Young inequality to measured groupoids

Use of non-commutative $L^p$-spaces in the analysis

Broader applicability in harmonic analysis

Abstract

The classical Hausdorff-Young inequality for locally compact abelian groups states that, for $1\le p\le 2$, the $L^p$-norm of a function dominates the $L^q$-norm of its Fourier transform, where $1/p+1/q=1$. By using the theory of non-commutative $L^p$-spaces and by reinterpreting the Fourier transform, R. Kunze (1958) [resp. M. Terp (1980)] extended this inequality to unimodular [resp. non-unimodular] groups. The analysis of the $L^p$-spaces of the von Neumann algebra of a measured groupoid provides a further extension of the Hausdorff-Young inequality to measured groupoids.