On the regularity of maximal operators

Abstract

Abstract. We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps W 1,p (R) × W 1,q (R) → W 1,r (R) with 1 < p, q < ∞ and r ≥ 1, boundedly and continuously. The same result holds on R n when r > 1. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versions.

On approximate differentiability of the maximal function

“…Let us also mention that the result of Kinnunen [10] has been applied and generalized by many authors ( [2], [3], [6], [7], [8], [11], [12], [14], [15], [16], [17], [18], [20], [21], [25]). …”

Regularity and continuity of commutators of the Hardy–Littlewood maximal function

“…The same conclusion also holds for by a simple modification of Kinnunen's arguments or [, Theorem 1]. Subsequently, Kinnunen's result was extended in a pretty much deeper way . Due to the lack of the sublinearity for the derivative of the maximal function, the continuity of for is certainly a nontrivial issue.…”

Regularity of Maximal Operators: Recent Progress and Some Open Problems

“…Kinnunen and Lindqvist [18] extended Theorem 2.1 to a local version of the maximal operator. In this setting one considers a domain Ω ⊂ R d , functions f ∈ W 1,p (Ω), and the maximal operator is taken over balls entirely contained in the domain Ω. Extensions of Theorem 2.1 to a multilinear setting are considered in the work of the author and Moreira [11] and by Liu and Wu [24], and a similar result in fractional Sobolev spaces is the subject of the work of Korry [20]. A fractional version of the Hardy-Littlewood maximal operator is considered in the paper [19] by Kinnunen and Saksman (we will return to this particular operator later on).…”