Convolution-type operators in grand Lorentz spaces

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.