A self-improving property of Riesz potentials in BMO

TL;DR

This paper establishes a self-improving property of Riesz potentials, showing that for non-negative functions, membership in BMO implies membership in a range of BMO-like spaces with different oscillation parameters.

Contribution

It proves an equivalence between BMO and BMO^β spaces for Riesz potentials, revealing a new self-improving property of these operators.

Findings

Riesz potentials of non-negative functions in BMO are also in BMO^β for β in (n-α,n]

The equivalence characterizes the oscillation behavior of Riesz potentials

The result links classical BMO with fractional oscillation spaces.

Abstract

In this paper we prove that for non-negative measurable functions $f$, \begin{align*} I_\alpha f \in BMO(\mathbb{R}^n) \text{ if and only if } I_\alpha f \in BMO^\beta(\mathbb{R}^n) \text{ for } \beta \in (n-\alpha,n]. \end{align*} Here $I_\alpha$ denotes the Riesz potential of order $\alpha$ and $BMO^\beta$ represents the space of functions of bounded $\beta$-dimensional mean oscillation.