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

N. V. Krylov

TL;DR

This work uncovers how Real Analysis techniques such as Hardy–Littlewood maximal functions, Fefferman–Stein inequalities, and Muckenhoupt weights can be systematically applied to nondivergence form elliptic and parabolic PDEs in Sobolev and Morrey–Sobolev spaces, including cases with singular first–order terms.A cohesive toolkit is developed: dyadic partitions and stopping times, maximal and sharp function controls, and weighted/mixed-norm frameworks, culminating in robust a priori estimates and solvability results for the Laplacian and elliptic operators with variable coefficients.Morrey–Sobolev spaces and Adams’s potential theory are integrated to handle lower-order singularities, with precise interpolation and embedding results that yield regularity for solutions in Morrey-type spaces and their weighted/mixed-norm extensions.The paper also provides a detailed treatment of Muckenhoupt weights, establishing weighted norm inequalities and reverse Hölder properties that underpin a broad class of PDE estimates in weighted settings, including parabolic/mixed-norm contexts.Overall, the work consolidates essential Real Analysis tools into a unified framework for analyzing second-order PDEs with singular coefficients, offering both classical results and new findings for variable-coefficient and Morrey-type problems.

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.

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

TL;DR

This work uncovers how Real Analysis techniques such as Hardy–Littlewood maximal functions, Fefferman–Stein inequalities, and Muckenhoupt weights can be systematically applied to nondivergence form elliptic and parabolic PDEs in Sobolev and Morrey–Sobolev spaces, including cases with singular first–order terms.A cohesive toolkit is developed: dyadic partitions and stopping times, maximal and sharp function controls, and weighted/mixed-norm frameworks, culminating in robust a priori estimates and solvability results for the Laplacian and elliptic operators with variable coefficients.Morrey–Sobolev spaces and Adams’s potential theory are integrated to handle lower-order singularities, with precise interpolation and embedding results that yield regularity for solutions in Morrey-type spaces and their weighted/mixed-norm extensions.The paper also provides a detailed treatment of Muckenhoupt weights, establishing weighted norm inequalities and reverse Hölder properties that underpin a broad class of PDE estimates in weighted settings, including parabolic/mixed-norm contexts.Overall, the work consolidates essential Real Analysis tools into a unified framework for analyzing second-order PDEs with singular coefficients, offering both classical results and new findings for variable-coefficient and Morrey-type problems.

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.

Paper Structure

This paper contains 32 sections, 149 theorems, 1031 equations.

Table of Contents

  1. Some tools from Real Analysis
  2. Partitions and stopping times
  3. Maximal and sharp functions
  4. Fefferman-Stein theorem
  5. Comparing the maximal and sharp functions
  6. Application to the Laplacian
  7. Elliptic equations with variable main coefficients
  8. Morrey-Sobolev spaces
  9. Adams's theorem
  10. Definition and simplest properties of Morrey-Sobolev spa-ces
  11. Interpolations and embeddings
  12. Solvability of the Laplace equation in Morrey-Sobolev spaces
  13. Muckenhoupt's weights
  14. Muckenhoupt's weights
  15. Application to the Laplacian
  16. ...and 17 more sections

Key Result

Lemma 1.1

(i) Let $f$ be Borel on $\mathbb{R}^{d}$, $f\geq0$, let $\tau$ be a stopping time, and let $A\in \mathcal{F}_{\tau}$. Then (ii) Let $f$ be Borel on $\mathbb{R}^{d}$, $f\geq0$, and let $\lambda>0$ be a constant. Assume that $f_{|n}(x)\to0$ as $n\to-\infty$ at any $x$. Then for $\tau=\tau_{\lambda}$ defined in (11.20.1) we have and for any $A\in\mathcal{F}_{\tau}$

Theorems & Definitions (217)

  • Lemma 1.1
  • Corollary 1.2
  • Remark 1.3
  • Theorem 1.4: Lebesgue
  • Remark 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Remark 1.8
  • Lemma 1.9
  • Theorem 1.10: Fefferman-Stein
  • ...and 207 more