Essentials of Real Analysis and Morrey-Sobolev spaces for second-order elliptic and parabolic PDEs with singular first-order coefficients

TL;DR

This paper explores how key results in Real Analysis, like maximal functions and weighted inequalities, can be applied to the study of elliptic and parabolic PDEs with singular coefficients in Sobolev and Morrey-Sobolev spaces, including new findings.

Contribution

It provides a concise introduction to applying Real Analysis tools to PDEs with singular coefficients, including new results on model equations like Laplace and heat equations.

Findings

Application of Hardy-Littlewood maximal function theorem to PDEs

Use of Muckenhoupt weights in Sobolev space theory

New results on elliptic and parabolic equations with singular first-order terms

Abstract

In recent years we witness growing interest in using Real Analysis methods and results in the theory of nondivergence form partial differential equations (PDEs) and the goal of this article is to give a brief and concise introduction into the applications of several results in Real Analysis to the theory of elliptic and parabolic equations in Sobolev and Morrey-Sobolev spaces. In particular, we concentrate on such results as Hardy-Littlewood maximal function theorem, Fefferman-Stein theorem, theory of Muckenhoupt weights, and Rubio de Francia extrapolation theorem and their role in Sobolev or Morrey-Sobolev space theory of parabolic equations with mixed norms. In our exposition we do not try to give the strongest known results for particular equations in particular spaces. We only show how the Real Analysis results, we present with all proofs, can be used in model cases such as the Laplace and the heat equations with singular first order terms. The only exception is the last section where we present new results.