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Characterizations of parabolic Muckenhoupt classes

Juha Kinnunen, Kim Myyryläinen

Abstract

This paper extends and complements the existing theory for the parabolic Muckenhoupt weights motivated by one-sided maximal functions and a doubly nonlinear parabolic partial differential equation of $p$-Laplace type. The main results include characterizations for the limiting parabolic $A_\infty$ and $A_1$ classes by applying an uncentered parabolic maximal function with a time lag. Several parabolic Calderón-Zygmund decompositions, covering and chaining arguments appear in the proofs.

Characterizations of parabolic Muckenhoupt classes

Abstract

This paper extends and complements the existing theory for the parabolic Muckenhoupt weights motivated by one-sided maximal functions and a doubly nonlinear parabolic partial differential equation of -Laplace type. The main results include characterizations for the limiting parabolic and classes by applying an uncentered parabolic maximal function with a time lag. Several parabolic Calderón-Zygmund decompositions, covering and chaining arguments appear in the proofs.
Paper Structure (11 sections, 20 theorems, 287 equations)

This paper contains 11 sections, 20 theorems, 287 equations.

Table of Contents

  1. Introduction
  2. Definition and properties of parabolic Muckenhoupt weights
  3. Time lag and distance between upper and lower parts
  4. Characterizations of parabolic $A_\infty$
  5. Quantitative measure condition
  6. Qualitative measure condition
  7. Sublevel measure condition
  8. Gurov--Reshetnyak condition
  9. Parabolic reverse Hölder inequality
  10. Weighted norm inequalities for parabolic maximal functions
  11. Factorization and characterization of parabolic Muckenhoupt weights

Key Result

Proposition 2.5

Let $0\leq\gamma<1$. A weight $w$ is in $A_1^+(\gamma)$ if and only if there exists a constant $C$ such that for almost every $(x,t) \in \mathbb{R}^{n+1}$. Moreover, we can choose $C = [w]_{A^+_1(\gamma)}$. The statement also holds for $A_1^-(\gamma)$ with $M^{\gamma+}$.

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 33 more