Regularity and Continuity of Local Multilinear Maximal Type Operators

Abstract

This paper will be devoted to study the regularity and continuity properties of the following local multilinear fractional type maximal operators,where Ω is a subdomain in R n , Ω c = R n \Ω and B(x, r) is the ball in R n centered at x with radius r. Several new pointwise estimates for the derivative of the local multilinear maximal function M0,Ω and the fractional maximal functions Mα,Ω (0 < α < mn) will be presented. These estimates will not only enable us to establish certain norm inequalities for these operators in Sobolev spaces, but also give us the opportunity to obtain the bounds of these operators on the Sobolev space with zero boundary values.

Regularity for commutators of the local multilinear fractional maximal operators

“…This result was later extended to the fractional version by Heikkinen et al [11]. Recently, Hart et al [10] established the following results.…”

“…Finally, we shall prove Corollaries 1.5-1.7 in Section 4. We would like to remark that the main ideas in our proofs are motivated by [10,11,14,21], but our methods and techniques are more subtle and complex than those of [10,11,14,21]. Particularly, some new techniques will be explored in our main proofs.…”

“…When m = , the operator M Ω (resp., M α,Ω ) is just the local (resp., fractional) maximal operator M Ω (resp., M α,Ω ). It was pointed out in [10] that…”

Regularity of Commutators of the One-Sided Hardy-Littlewood Maximal Functions

“…This problem was addressed by Luiro [7] in the affirmative and was later extended to the local version in [8] and the multisublinear version in [4,9]. Other works on the regularity of maximal operators can be consulted in [10,11]. Since the map M :…”

Gradient bounds for radial maximal functions

“…In the seminal paper [14], Kinnunen studied the action of the Hardy-Littlewood maximal operator on Sobolev functions, giving an elegant proof that M : W 1,p (R d ) → W 1,p (R d ) is bounded for 1 < p ≤ ∞. This work paved the way for several interesting contributions to the regularity theory of maximal operators over the past two decades, with interesting connections to potential theory and partial differential equations, see for instance [1,3,4,6,7,8,11,13,15,16,18,19,22,23,21,24,25,26]. One of the longstanding problems in this field is concerned with the regularity at the endpoint p = 1.…”