Abstract
In this paper, the regularity properties of two classes of commutators of the one-sided Hardy-Littlewood maximal functions and their fractional variants are investigated. Some new bounds for the derivatives of the above commutators and the boundedness and continuity for the above commutators on the Sobolev spaces will be presented. The corresponding results for the discrete analogues are also considered.
Highlights
The regularity theory of maximal operators has been the subject of many recent articles in harmonic analysis
One of the driving questions in this theory is whether a given maximal operator improves, preserves, or destroys a priori regularity of an initial datum f
The question was first studied by Kinnunen [1], who showed that the usual centered HardyLittlewood maximal function M is bounded on the first order Sobolev spaces W1,pðRdÞ for all 1 < p ≤ ∞
Summary
The regularity theory of maximal operators has been the subject of many recent articles in harmonic analysis. The main motivations of this work extend Theorem 1 to a one-sided setting and investigate the regularity properties of the discrete analogue for commutators of the one-sided Hardy-Littlewood maximal functions and their fractional variants. It is currently unknown whether inequality (29) holds for the discrete centered fractional maximal operator It was pointed out in [30] that both the maps f ↦ ðMβ f Þ′ and f ↦ ðM~ β f Þ′ (for 0 ≤ β < 1) are bounded and continuous from l1ðZÞ to l1ðZÞ. Liu and Mao [6] studied the regularity of the discrete one-sided Hardy-Littlewood maximal operators and proved them. The letter Cα,β denotes the positive constants that depend on the parameters α, β
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