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arXiv · 2409.17918

Classical inequalities for all Fourier matrix coefficients of $\mathrm{SL}(2,\mathbb{R})$ and their applications

Vishvesh Kumar·Tapendu Rana·Michael Ruzhansky
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Abstract

In this article, we establish three fundamental Fourier inequalities: the Hausdorff-Young inequality, the Paley inequality, and the Hausdorff-Young-Paley inequality for $(l, n)$-type functions on $\mathrm{SL}(2,\mathbb{R})$. Utilizing these inequalities, we demonstrate the $L^p$-$L^q$ boundedness of $(l, n)$-type Fourier multipliers on $\mathrm{SL}(2,\mathbb{R})$. Furthermore, we explore applications related to the $L^p$-$L^q$ estimates of the heat kernel of the Casimir element on $\mathrm{SL}(2,\mathbb{R})$ and address the global well-posedness of certain parabolic and hyperbolic nonlinear equations.

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Vishvesh Kumar, Tapendu Rana, Michael Ruzhansky. 2024-09-26. Classical inequalities for all Fourier matrix coefficients of $\mathrm{SL}(2,\mathbb{R})$ and their applications. https://arxiv.org/abs/2409.17918

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