$L^p$-$L^q$ boundedness of $(k, a)$-Fourier multipliers with applications to Nonlinear equations

TL;DR

This paper establishes $L^p$-$L^q$ boundedness for $(k,a)$-generalized Fourier multipliers using inequalities like Paley and Hausdorff-Young-Paley, and applies these results to analyze nonlinear PDEs.

Contribution

It proves new $L^p$-$L^q$ bounds for $(k,a)$-generalized Fourier multipliers and demonstrates their application to nonlinear equations.

Findings

Proved Paley inequality for $(k,a)$-Fourier transform.

Established Hausdorff-Young-Paley inequality for the transform.

Applied results to study well-posedness of nonlinear PDEs.

Abstract

The $(k,a)$-generalised Fourier transform is the unitary operator defined using the $a$-deformed Dunkl harmonic oscillator.The main aim of this paper is to prove $L^p$-$L^q$ boundedness of $(k, a)$-generalised Fourier multipliers. To show the boundedness we first establish Paley inequality and Hausdorff-Young-Paley inequality for $(k, a)$-generalised Fourier transform. We also demonstrate applications of obtained results to study the well-posedness of nonlinear partial differential equations.