The Fourier transform in variable exponent Lebesgue spaces

2025/06/10 by Kowacs, André Pedroso, de Moraes, Wagner Augusto Almeida
#46B04 #46E30. Secondary: 26A42 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary: 42A38

paper · doi:10.48550/arxiv.2506.08891

Abstract

In this work we define a Fourier transform for each f∈ L^p(⋅)(ℝ), for a large class of exponent functions p(⋅), as the distributional derivative of a Hölder continuous function. A norm is defined in the space of such Fourier transforms so that it is isometrically isomorphic to L^p(⋅)(ℝ). We also prove several properties of this Fourier transform, such as inversion in norm and an exchange theorem.

Citations

• On variable Lebesgue spaces and generalized nonlinear heat equations
• Lebesgue spaces with variable exponent: some applications to the Navier-Stokes equations
• The Fourier transform in Lebesgue spaces
• New Characterizations of Musielak-Orlicz-Sobolev Spaces via Sharp Ball Averaging Functions
• The distributional Denjoy integral

Related

• The Fourier transform in Lebesgue spaces
• Fourier inequalities in variable Lebesgue spaces
• The Fourier transform with Henstock--Kurzweil and continuous primitive integrals
• The Fourier Transform of Anisotropic Hardy Spaces with Variable Exponents and Their Applications
• Mapping properties of the Fourier transform in spaces with dominating mixed smoothness