A note on the Muckenhoupt class $A_1(\mathbb{R})$

TL;DR

This paper presents a direct proof of the optimal reverse Hölder inequality for weights in the Muckenhoupt A1 class on the interval (0,1), clarifying their integrability properties.

Contribution

It offers the first direct proof of the sharp reverse Hölder inequality for A1 weights on (0,1), improving understanding of their behavior.

Findings

Established the best possible reverse Hölder inequality for A1 weights

Demonstrated the inequality's sharpness and optimality

Enhanced theoretical understanding of A1 weight properties

Abstract

We provide a direct proof of the best possible reverse Holder inequality satisfied by every weight defined on the interval $(0,1)$ with A1-constant equal to c > 1.