Parabolic weak porosity and parabolic Muckenhoupt distance functions

TL;DR

This paper introduces parabolic weak porosity to characterize parabolic Muckenhoupt $A_1$ weights with time-lag, linking geometric properties of sets to weight classes via a novel stopping time approach.

Contribution

It establishes a characterization of parabolic $A_1$ weights using parabolic weak porosity and develops new techniques involving stopping times and set translation.

Findings

A set is parabolic weakly porous iff its distance function to a negative power belongs to $A_1$.

The paper applies a novel stopping time argument with translation and doubling results.

Provides a geometric-analytic characterization of parabolic $A_1$ weights.

Abstract

We develop the parabolic weak porosity to characterize the parabolic Muckenhoupt $A_1$ weights with time-lag. Our main result shows that a nonempty closed set is parabolic weakly porous if and only if the parabolic distance function of the set to a negative power is in the parabolic Muckenhoupt $A_1$ class. We apply a novel stopping time argument in combination with the translation and doubling results for the parabolic weakly porous sets.