Characterizations of BMO with Hausdorff content

Abstract

Let $H_{\infty }^{\delta }$ be the Hausdorff content on $R^n$ of order $\delta \in (0,n]$. For the space $\mathrm{BMO}\left(H_{\infty }^{\delta }\right)$ of bounded mean oscillation that is defined with respect to $H_{\infty }^{\delta }$, the authors establish its equivalent characterizations via replacing the $L^1\left(H_{\infty }^{\delta }\right)$-integrability in its definition by the Luxemburg type $L^{\phi }\left(H_{\infty }^{\delta }\right)$-integrability, where φ can be either a convex or a concave nonnegative function. Typical examples of φ include $\phi \left(t\right)=t^p$, $\phi \left(t\right)=e^{pt}-1$ and $\phi \left(t\right)={\log }_{+}\circ \cdots \circ {\log }_{+}t$, where $p\in (0,\infty )$ and ${\log }_{+}t:=\max \{\log t,0\}$.