$\beta$-dimensional sharp maximal function and applications

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.