John-Nirenberg inequalities for parabolic BMO

TL;DR

This paper extends John-Nirenberg inequalities to parabolic BMO spaces associated with nonlinear PDEs, establishing exponential decay estimates and boundary conditions in the natural PDE geometry.

Contribution

It introduces a parabolic version of John-Nirenberg inequalities, including boundary conditions and median-based variants, for functions related to nonlinear parabolic PDEs.

Findings

Established parabolic John-Nirenberg inequalities in natural PDE geometry

Proved quasihyperbolic boundary condition as necessary and sufficient

Demonstrated median-based inequalities coincide with original class

Abstract

We discuss a parabolic version of the space of functions of bounded mean oscillation related to a doubly nonlinear parabolic partial differential equation. Parabolic John-Nirenberg inequalities, which give exponential decay estimates for the oscillation of a function, are shown in the natural geometry of the partial differential equation. Chaining arguments are applied to change the time lag in the parabolic John-Nirenberg inequality. We also show that the quasihyperbolic boundary condition is a necessary and sufficient condition for a global parabolic John-Nirenberg inequality. Moreover, we consider John-Nirenberg inequalities with medians instead of integral averages and show that this approach gives the same class of functions as the original definition.