Paley inequality for the Weyl transform and its applications

TL;DR

This paper extends classical inequalities like Paley, Hardy-Littlewood, and Hausdorff-Young to the Weyl transform, establishing boundedness results and multiplier theorems with vector-valued considerations.

Contribution

It introduces new versions of Paley inequality for the Weyl transform and explores their applications to multiplier theorems and vector-valued inequalities.

Findings

Established several Paley inequalities for the Weyl transform.

Proved $L^p$-$L^q$ boundedness of Weyl multipliers.

Extended Hardy-Littlewood and Hausdorff-Young inequalities to this setting.

Abstract

In this paper, we prove several versions of the classical Paley inequality for the Weyl transform. As an application, we discuss $L^p$-$L^q$ boundedness of the Weyl multipliers and prove a version of the H\"ormander's multiplier theorem. We also prove Hardy-Littlewood inequality. Finally, we study vector-valued versions of these inequalities. In particular, we consider the inequalities of Paley, Hausdorff-Young, and Hardy-Littlewood and their relations.