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

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.

20 pages. arXiv admin note: text overlap with arXiv:2108.01146

References in corpus (5)

• $L^p$-$L^q$ boundedness of pseudo-differential operators on smooth manifolds and its applications to nonlinear equations
• Hardy inequalities for fractional $(k,a)$-generalized harmonic oscillator
• Hardy-Littlewood inequality and $L^p$-$L^q$ Fourier multipliers on compact hypergroups
• $L^p-L^q$ boundedness of Fourier multipliers associated with the anharmonic Oscillator
• $L^p-L^q$ estimates for the solution of the Dunkl wave equation

Cited by in corpus (5)

• Fourier multipliers and their applications to PDE on the quantum Euclidean space
• A note on $L^p$-$L^q$ boundedness of Fourier multipliers on noncommutative spaces
• $L^p$-$L^q$ boundedness of Fourier multipliers on Fundamental domains of Lattices in ${\mathbb{R}}^d$
• $L^p-L^q$ boundedness of Fourier multipliers on quantum Euclidean spaces
• Hörmander type Fourier multiplier theorem and Nikolskii inequality on quantum tori, and applications