Convolution-type operators in grand Lorentz spaces
Erlan D. Nursultanov, Humberto Rafeiro, Durvudkhan Suragan
TL;DR
This work introduces grand Lorentz spaces $GL_{p,q}^\theta(\Omega)$ as a refined aggregation between Lorentz-Karamata and existing grand Lorentz spaces. It develops a comprehensive operator theory in this framework by proving O'Neil-type and Young-type convolution inequalities and a Hardy-Littlewood-Sobolev-type bound through power-logarithmic Riesz operators. The paper also establishes real interpolation results and a robust Köthe duality theory, showing $GL_{p,q}^\theta(\Omega)=(GL_{p',q'}^{-\theta}(\Omega))'$ and clarifying duality with grand Lebesgue spaces, thereby providing a versatile toolkit for endpoint estimates in critical cases. These contributions enhance the functional-analytic apparatus for analyzing convolution-type operators in refined scale spaces with precise control near endpoints.
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 Köthe duality for grand Lorentz spaces, from which we obtain a new Köthe dual space theorem in grand Lebesgue spaces.