A self-improving property of Riesz potentials in BMO

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.