A self-improving property of Riesz potentials in BMO

Abstract

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