Sharp transference principle for $\mathrm{BMO}$ and $A_p$

TL;DR

This paper establishes a transference principle showing that key inequalities and constants in BMO spaces and Muckenhoupt weights are equivalent across different domains, unifying their analysis.

Contribution

It introduces a transference principle that equates optimization problems and sharp constants for BMO spaces and Muckenhoupt weights across various settings.

Findings

Sharp constants in John--Nirenberg inequalities are identical on the circle, interval, and line.

The transference principle applies to the Reverse H"older inequality for Muckenhoupt weights.

Unified understanding of inequalities across different domains.

Abstract

We provide a version of the transference principle. It says that certain optimization problems for functions on the circle, the interval, and the line have the same answers. In particular, we show that the sharp constants in the John--Nirenberg inequalities for naturally defined $\mathrm{BMO}$-spaces on the circle, the interval, and the line coincide. The same principle holds true for the Reverse H\"older inequality for Muckenhoupt weights.