Fourier series in weighted Lorentz spaces

TL;DR

This paper investigates the boundedness of the Fourier coefficient map between weighted Lorentz spaces, providing necessary and sufficient conditions on weights and extending known inequalities to new Lorentz-Zygmund spaces.

Contribution

It offers new characterizations of weight conditions for boundedness and introduces direct analogues of Lorentz space inequalities for the Fourier transform.

Findings

Characterization of weights for bounded Fourier coefficient map

Extension of Lorentz space inequalities to Fourier transform

Applications to functions in LlogL and Lorentz-Zygmund spaces

Abstract

The Fourier coefficient map is considered as an operator from a weighted Lorentz space on the circle to a weighted Lorentz sequence space. For a large range of Lorentz indices, necessary and sufficient conditions on the weights are given for the map to be bounded. In addition, new direct analogues are given for known weighted Lorentz space inequalities for the Fourier transform. Applications are given that involve Fourier coefficients of functions in LlogL and more general Lorentz-Zygmund spaces.