Convolution-type operators in grand Lorentz spaces

TL;DR

This paper introduces a new grand Lorentz space bridging existing spaces, proves key inequalities within it, and explores duality properties, advancing the mathematical framework for critical cases.

Contribution

It defines a novel grand Lorentz space, establishes fundamental inequalities, and analyzes duality, extending the theory of Lorentz and Lebesgue spaces.

Findings

Proved Young's and O'Neil's inequalities in the new space

Derived a Hardy-Littlewood-Sobolev-type inequality

Established a new K"othe dual space theorem

Abstract

We introduce and study a novel grand Lorentz space-that we believe is appropriate for critical cases-that lies "between" the Lorentz-Karamata space and the recently defined grand Lorentz space from [1]. We prove both Young's and O'Neil's inequalities in the newly introduced grand Lorentz spaces, which allows us to derive a Hardy-Littlewood-Sobolev-type inequality. We also discuss K\"othe duality for grand Lorentz spaces, from which we obtain a new K\"othe dual space theorem in grand Lebesgue spaces.