β-dimensional sharp maximal function and applications

In this paper, we study β-dimensional sharp maximal operator defined as {align*} M# _β f(x) := Q infc ∈ R χQ(x) 1/(Q)^β ∫_Q |f-c| \; d Hβ_∞, {align*} where the supremum is taken over all cubes in Rᵈ with sides pararell to the coordinate axes, (Q) is the length side of Q and Hβ_∞ is the Hausdorff content. In particular, we prove Fefferman-Stein inequality for M# _β f by giving a good lambda estimate for β-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.