Weak-$L^\infty$ and BMO with Hausdorff Content

Abstract

The space of weak-$L^{\infty}$ on measure space was introduced by Bennett et al. (Ann. of Math., 113(1981), 601-611) and proved to contain the space of $L^\infty$ and possess the appropriate interpolation property. In this paper, we are aimed to continue this line of research on the Euclidean space $\mathbb R^n$ endowed with Hausdorff content, which is a low-dimensional outer measure on $\mathbb{R}^n$, and fails to be additive or absolutely continuous with respect to Lebesgue measures. But low-dimensional subsets, such as fractals, of $\mathbb{R}^n$ are Hausdorff capacitable. Precisely, in the setting of Hausdorff content, an analogue of weak-$L^\infty$ is defined and proved to possess an improved Marcinkiewicz interpolation theorem. Moreover, we show that the space of functions with bounded mean oscillation associated to the Hausdorff content is embedded into this new weak-$L^\infty$ space by establishing a key estimate for the distribution via a packing condition. As a by-product, the boundedness of the Hardy-Littlewood operator on the space of weak-$L^{\infty}$ is established.