Weak differentiability for fractional maximal functions of general L functions on domains

Abstract

Let Ω ⊂ R n be bounded a domain. We prove under certain structural assumptions that the fractional maximal operator relative to Ω maps L p (Ω) → W 1,p (Ω) for all p > 1, when the smoothness index α ≥ 1. In particular, the results are valid in the range p ∈ (1, n/(n − 1)] that was previously unknown. As an application, we prove an endpoint regularity result in the domain setting.

Regularity for commutators of the local multilinear fractional maximal operators

“…By Riesz theorem, there exists a subsequence {φ ,j k } ∞ k= ⊂ {φ ,j } ∞ j= such that φ ,j k (x) → b (x) as k → ∞ for almost every x ∈ Ω. By the arguments similar to those used to the proof of Claim 2 of Lemma 3.4, we have 29) for almost every x ∈ Ω. Same arguments to those in deriving (3.17) will give…”

Variation of the uncentered maximal characteristic function

“…Also related topics for various exponents 1 Ä p Ä 1 have been studied, such as the continuity of the maximal operator in Sobolev spaces [5] and bounds for the gradient of other maximal operators, such as fractional, convolution, discrete, local and bilinear maximal operators [6,10,11,16,19,20,23,25].…”