New directions in harmonic analysis on L1

Daniel Spector*

*Corresponding author for this work

Research output: Contribution to journalArticle

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 do not understand about the action of integral operators on functions. This is no more apparent than in the L1 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 present the state of the art of what is known, and to survey some open problems on the frontier of research in the area.

Original languageEnglish
Article number111685
JournalNonlinear Analysis, Theory, Methods and Applications
Volume192
DOIs
StatePublished - Mar 2020

Keywords

  • L estimates
  • Riesz potentials
  • Sobolev inequalities

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