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arXiv · 2406.06268

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

Pratyoosh Kumar·Sanjoy Pusti·Tapendu Rana·Mandeep Singh
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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.

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Pratyoosh Kumar, Sanjoy Pusti, Tapendu Rana, Mandeep Singh. 2024-06-10. Weighted Fourier inequalities and application of restriction theorems on rank one Riemannian symmetric spaces of noncompact type. https://arxiv.org/abs/2406.06268

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