${\hat{I}}^2$-dimensional sharp maximal function and applications

Abstract

In this paper, we study ${\hat{I}}^2$-dimensional sharp maximal operator defined as \begin{align*} \mathcal{M}^{\#} _βf(x) := \sup_{Q} \inf_{c \in \mathbb{R}} χ_{Q}(x) \frac{1}{\ell(Q)^β} \int_Q |f-c| \; d \mathcal{H}^β_\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 ${\mathcal{H}}_{\infty }^2$ is the Hausdorff content. In particular, we prove Fefferman-Stein inequality for ${\mathcal{M}}_{\hat{I}}^{2}f$ by giving a good lambda estimate for ${\hat{I}}^2$-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.

30 pages