Extrapolation in general quasi-Banach function spaces

TL;DR

This paper extends the Rubio de Francia extrapolation theorem to general quasi-Banach function spaces, providing new bounds and applications for operators like the Hardy-Littlewood maximal operator, Riesz potential, and Bilinear Hilbert transform.

Contribution

It introduces off-diagonal, limited range, multilinear, vector-valued, and two-weight extrapolation results in quasi-Banach spaces, extending classical theorems and methods.

Findings

Proved new extrapolation theorems in quasi-Banach spaces.

Established bounds for Hardy-Littlewood maximal operator in various spaces.

Applied results to Riesz potential and Bilinear Hilbert transform.

Abstract

In this paper we prove off-diagonal, limited range, multilinear, vector-valued, and two-weight versions of the Rubio de Francia extrapolation theorem in general quasi-Banach function spaces. We prove mapping properties of the generalization of the Hardy-Littlewood maximal operator to very general bases that includes a method to obtain self-improvement results that are sharp with respect to its operator norm. Furthermore, we prove bounds for the Hardy-Littlewood maximal operator in weighted Lorentz, variable Lebesgue, and Morrey spaces, and recover and extend several extrapolation theorems in the literature. Finally, we provide an application of our results to the Riesz potential and the Bilinear Hilbert transform.