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One-Sided and Parabolic BLO Spaces with Time Lag and Their Applications to Muckenhoupt $A_1$ Weights and Doubly Nonlinear Parabolic Equations

Weiyi Kong, Dachun Yang, Wen Yuan

TL;DR

The paper develops a comprehensive endpoint theory for one-sided BLO spaces and their parabolic analogues. It provides a complete characterization of $\mathrm{BLO}^+(\mathbb{R})$ via the one-sided Muckenhoupt class $A_1^+(\mathbb{R})$ and a one-sided John–Nirenberg inequality, and proves a Coifman–Rochberg type decomposition together with Bennett-type representations linking BLO$^+$ to $\mathrm{BMO}^+$. The authors extend these results to the parabolic setting by introducing $\mathrm{PBLO}_\gamma^-(\mathbb{R}^{n+1})$ and establishing lag-independent characterizations, Bennett-type relations, and a CR decomposition, then connect the theory to doubly nonlinear parabolic equations and parabolic porosity concepts. They also derive geometric implications, showing that the negative parabolic distance to a set belongs to $\mathrm{PBLO}_\gamma^-(\mathbb{R}^{n+1})$ under a parabolic weak porosity condition, thereby linking harmonic-analytic spaces with PDE regularity and geometric measure properties.

Abstract

In this article, we first introduce the one-sided BLO space $\mathrm{BLO}^+(\mathbb{R})$ and characterize it, respectively, in terms of the one-sided Muckenhoupt class $A_1^+(\mathbb{R})$ and the one-sided John--Nirenberg inequality. Using these, we establish the Coifman--Rochberg type decomposition of $\mathrm{BLO}^+(\mathbb{R})$ functions and show that $\mathrm{BLO}^+(\mathbb{R})$ is independent of the distance between the two intervals, which further induces the characterization of this space in terms of the one-sided BMO space $\mathrm{BMO}^+(\mathbb{R})$ (the Bennett type lemma). As applications, we prove that any $\mathrm{BMO}^+(\mathbb{R})$ function can split into the sum of two $\mathrm{BLO}^+(\mathbb{R})$ functions and we provide an explicit description of the distance from $\mathrm{BLO}^+(\mathbb{R})$ functions to $L^\infty(\mathbb{R})$. Finally, as a higher-dimensional analogue we introduce the parabolic BLO space $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$ with time lag, and we extend all the above one-dimensional results to $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$; furthermore, as applications, we not only establish the relationships between $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$ and the solutions of doubly nonlinear parabolic equations, but also provide a necessary condition for the negative logarithm of the parabolic distance function to belong to $\mathrm{PBLO}_γ^-(\mathbb{R}^{n+1})$ in terms of the weak porosity of the set.

One-Sided and Parabolic BLO Spaces with Time Lag and Their Applications to Muckenhoupt $A_1$ Weights and Doubly Nonlinear Parabolic Equations

TL;DR

The paper develops a comprehensive endpoint theory for one-sided BLO spaces and their parabolic analogues. It provides a complete characterization of via the one-sided Muckenhoupt class and a one-sided John–Nirenberg inequality, and proves a Coifman–Rochberg type decomposition together with Bennett-type representations linking BLO to . The authors extend these results to the parabolic setting by introducing and establishing lag-independent characterizations, Bennett-type relations, and a CR decomposition, then connect the theory to doubly nonlinear parabolic equations and parabolic porosity concepts. They also derive geometric implications, showing that the negative parabolic distance to a set belongs to under a parabolic weak porosity condition, thereby linking harmonic-analytic spaces with PDE regularity and geometric measure properties.

Abstract

In this article, we first introduce the one-sided BLO space and characterize it, respectively, in terms of the one-sided Muckenhoupt class and the one-sided John--Nirenberg inequality. Using these, we establish the Coifman--Rochberg type decomposition of functions and show that is independent of the distance between the two intervals, which further induces the characterization of this space in terms of the one-sided BMO space (the Bennett type lemma). As applications, we prove that any function can split into the sum of two functions and we provide an explicit description of the distance from functions to . Finally, as a higher-dimensional analogue we introduce the parabolic BLO space with time lag, and we extend all the above one-dimensional results to ; furthermore, as applications, we not only establish the relationships between and the solutions of doubly nonlinear parabolic equations, but also provide a necessary condition for the negative logarithm of the parabolic distance function to belong to in terms of the weak porosity of the set.
Paper Structure (10 sections, 31 theorems, 195 equations)

This paper contains 10 sections, 31 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Characterizations of $\mathrm{BLO}^+(\mathbb{R})$ in Terms of $A_1^+(\mathbb{R})$ and John--Nirenberg Inequality
  3. Bennett Type Characterization of $\mathrm{BLO}^+(\mathbb{R})$ in Terms of $\mathrm{BMO}^+(\mathbb{R})$
  4. Coifman--Rochberg Type Decomposition of $\mathrm{BLO}^+(\mathbb{R})$ Functions and Distances of Functions in $\mathrm{BLO}^+(\mathbb{R})$ to ${L^\infty(\mathbb{R})}$
  5. Higher-Dimensional Extensions and Applications to Doubly Nonlinear Parabolic Equations
  6. Characterizations of $\mathrm{PBLO}_\gamma^-(\mathbb{R}^{n+1})$ in Terms of $A_1^+(\gamma)$
  7. Bennett Type Characterization of $\mathrm{PBLO}_\gamma^-(\mathbb{R}^{n+1})$ in Terms of $\mathrm{PBMO}_\gamma^-(\mathbb{R}^{n+1})$
  8. Coifman--Rochberg Type Decomposition of $\mathrm{PBLO}_\gamma^-(\mathbb{R}^{n+1})$
  9. Applications to Doubly Nonlinear Parabolic Equations
  10. Connections with Weakly Porous Sets

Key Result

Lemma 2.1

Let $q\in(1,\infty)$ and $\omega$ be a weight. Then $\omega\in A_q^+(\mathbb{R})$ if and only if $\omega\in A_\infty^+(\mathbb{R})$ and $\omega^\frac{1}{1-q}\in A_\infty^-(\mathbb{R})$. Moreover,

Theorems & Definitions (61)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • proof : Proof of Theorem \ref{['BLO+ and A1+']}
  • Remark 2.6
  • Theorem 2.7
  • ...and 51 more