In this paper, we study $\beta$-dimensional sharp maximal operator defined as \begin{ align*} \mathcal{ M}^{ \#} _\beta f(x) := \sup_{ Q} \inf_{ c \in \mathbb{ R}} \chi_{ Q}(x) \frac{ 1}{ \ell(Q)^\beta} \int_Q |f-c| \; d \mathcal{ H}^{ \beta}_\infty, \end{ align*} where the supremum is taken over all cubes in $\mathbb{ R}^d$ with sides pararell to the coordinate axes, $\ell(Q)$ is the length side of $Q$ and $\mathcal{ H}^{ \beta}_\infty$ is the Hausdorff content. In particular, we prove Fefferman-Stein inequality for $\mathcal{ M}^{ \#} _\beta f$ by giving a good lambda estimate for $\beta$-dimensional sharp maximal operator in the context of Hausdorff content. Additionally, we prove the Muckenhoupt-Wheeden inequality in this framework by establishing a good lambda inequality of independent interest.
Categories: math.FA