Weighted Fourier inequalities and application of restriction theorems on rank one Riemannian symmetric spaces of noncompact type

TL;DR

This paper establishes weighted Fourier inequalities on rank one noncompact symmetric spaces, using restriction theorems and spherical functions, and applies these results to polynomial and exponential weights.

Contribution

It provides necessary and sufficient conditions for weighted Fourier inequalities on rank one symmetric spaces, extending classical results with new applications.

Findings

Necessary and sufficient conditions for weighted inequalities

Application of restriction theorems on symmetric spaces

Derivation of inequalities with polynomial and exponential weights

Abstract

This article explores weighted $(L^p, L^q)$ inequalities for the Fourier transform in rank one Riemannian symmetric spaces of noncompact type. We establish both necessary and sufficient conditions for these inequalities to hold. To prove the weighted Fourier inequalities, we apply restriction theorems on symmetric spaces and utilize Calder{\'o}n's estimate for sublinear operators. While establishing the necessary conditions, we demonstrate that Harish-Chandra's elementary spherical functions play a crucial role in this setting. Furthermore, we apply our findings to derive Fourier inequalities with polynomial and exponential weights.