β-dimensional sharp maximal function and applications

2024/07/05 by Chen, You-Wei Benson, Claros, Alejandro
#42B25 #42B35 #46E35 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2407.04456

Abstract

In this paper, we study β-dimensional sharp maximal operator defined as M^# _βf(x) := sup_Q inf_{c ∈ ℝ} χ_Q(x) (1)/(ℓ(Q)^β) ∫_Q |f-c| d H^β_∞, where the supremum is taken over all cubes in ℝ^d with sides pararell to the coordinate axes, ℓ(Q) is the length side of Q and H^β_∞ is the Hausdorff content. In particular, we prove Fefferman-Stein inequality for M^# _βf 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.

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