  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.
