Parabolic John-Nirenberg spaces with time lag

TL;DR

This paper introduces a parabolic version of John-Nirenberg spaces, establishing inequalities that provide weak type estimates for functions' oscillations within a parabolic geometry incorporating a time lag.

Contribution

It generalizes the classical John-Nirenberg space to a parabolic setting with time lag, using Calderón-Zygmund decomposition and good lambda techniques.

Findings

Established parabolic John-Nirenberg inequalities with weak type estimates.

Developed a method to change the time lag in inequalities using chaining arguments.

Extended the theory of functions of bounded mean oscillation to parabolic geometries.

Abstract

We introduce a parabolic version of the so-called John-Nirenberg space that is a generalization of functions of parabolic bounded mean oscillation. Parabolic John-Nirenberg inequalities, which give weak type estimates for the oscillation of a function, are shown in the setting of the parabolic geometry with a time lag. Our arguments are based on a parabolic Calder\'{o}n-Zygmund decomposition and a good lambda estimate. Chaining arguments are applied to change the time lag in the parabolic John-Nirenberg inequality.