Nonstandard Calder\'on-type theorems

TL;DR

This paper proves Calderón-type theorems for operators on nonstandard Lorentz spaces, focusing on endpoint cases and space improvements, advancing the understanding of operator boundedness in these complex function spaces.

Contribution

It establishes new Calderón-type theorems for operators on nonstandard Lorentz spaces, particularly at endpoint cases, and explores associated space improvements.

Findings

Proved boundedness of operators on nonstandard Lorentz spaces.

Identified conditions for space improvements related to these operators.

Focused on cases where q0=q1 and q1=∞.

Abstract

We establish Calder\'on-type theorems for operators bounded on nonstandard end-point Lorentz spaces \begin{equation*} T\colon L^{p_0, q_0}\to L^{p_1, q_1}\quad\text{and}\quad T\colon L^{q, 1}\to L^\infty \end{equation*} and the improvement of target spaces which is intimately connected with this. The emphasis will be placed on the cases $q_0=q_1$ and $q_1=\infty$.