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

TL;DR

This paper introduces and characterizes one-sided BLO spaces and their parabolic analogues with time lag, explores their relationships with Muckenhoupt weights and BMO spaces, and applies these to doubly nonlinear parabolic equations.

Contribution

It develops a comprehensive theory of one-sided BLO and parabolic BLO spaces, including characterizations, decompositions, and applications to nonlinear PDEs and weight classes.

Findings

Characterization of $ ext{BLO}^+( r)$ via $A_1^+( r)$ and John--Nirenberg inequality.

Decomposition of $ ext{BMO}^+( r)$ into $ ext{BLO}^+( r)$ functions.

Extension of results to parabolic BLO spaces with applications to nonlinear parabolic equations.

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 $\mathrm{PBLO}_\gamma^-(\mathbb{R}^{n+1})$ with time lag, and we extend all the above one-dimensional results to $\mathrm{PBLO}_\gamma^-(\mathbb{R}^{n+1})$; furthermore, as applications, we not only establish the relationships between $\mathrm{PBLO}_\gamma^-(\mathbb{R}^{n+1})$ 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 $\mathrm{PBLO}_\gamma^-(\mathbb{R}^{n+1})$ in terms of the weak porosity of the set.