New Directions in Harmonic Analysis on $L^1$

TL;DR

This paper surveys recent advances and open problems in harmonic analysis on $L^1$, focusing on integral operators, Sobolev inequalities, and the classical Poisson equation, highlighting gaps in understanding especially for vector-valued functions.

Contribution

It provides a survey of recent optimal inequalities in the $L^1$ setting and discusses overlooked estimates that open new research directions.

Findings

Recent optimal inequalities for scalar functions on $L^1$ and Lorentz scales.

Incomplete understanding of vector-valued function estimates.

Identification of open problems in harmonic analysis on $L^1$.

Abstract

The study of what we now call Sobolev inequalities has been studied for almost a century in various forms, while it has been eighty years since Sobolev's seminal mathematical contributions. Yet there are still things we don't understand about the action of integral operators on functions. This is no more apparent than in the $L^1$ setting, where only recently have optimal inequalities been obtained on the Lebesgue and Lorentz scale for scalar functions, while the full resolution of similar estimates for vector-valued functions is incomplete. The purpose of this paper is to discuss how some often overlooked estimates for the classical Poisson equation give an entry into these questions, to the present state of the art of what is known, and to survey some open problems on the frontier of research in the area.