On the regularity of the Hardy-Littlewood maximal operator on subdomains of ℝⁿ

Abstract

We establish the continuity of the Hardy-Littlewood maximal operator on W1,p(Ω), where Ω ⊂ ℝn is an arbitrary subdomain and 1 < p < ∞. Moreover, boundedness and continuity of the same operator is proved on the Triebel-Lizorkin spaces Fps,q (Ω) for 1 < p,q < ∞ and 0 < s < 1.

Regularity and Continuity of Local Multilinear Maximal Type Operators

“…(a) Theorem 2.1 extends (i) of some results from [19,33,15], and Theorems 2.2-2.4 extend results from [16], which correspond to the case m = Recall that the Sobolev space with zero boundary values, denoted by W 1,p 0 (Ω) with 1 ≤ p < ∞, is defined as the completion of C ∞ 0 (Ω) with respect to the Sobolev norm. In 1998, Kinnunen and Lindqvist [19] observed that M Ω :…”

“…Lemma 5.4 (Lemma 2.11, [33]). Let A j ⊂ R n be measurable sets and let h k ∈ R n such that |h k | → 0 when k → ∞.…”

“…A complete answer for this continuity problem was first given by Lurio in [32]. Then, similar results of continuity were soon extended to a kind of bilinear maximal functions [7] and to local maximal functions [33]. To emphasize the need to address such subtle details, it should be noted that bounded non-sublinear operators need not to be continuous (see [4] for example).…”

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

“…More generally, Korry [19] proved that M is bounded on the inhomogeneous Triebel-Lizorkin spaces , ( ℝ ) and inhomogeneous Besov spaces , ( ℝ ) for 0 < < 1 and 1 < , < ∞. Later on, Luiro [29] established the continuity of on , ( ℝ ) for 0 < < 1 and 1 < , < ∞. Recently, Liu and Wu [25] extended the above results to the maximal operators associated with polynomial mappings.…”

A Note on the Regularity of the Two-Dimensional One-Sided Hardy-Littlewood Maximal Function

“…This problem was addressed by Luiro [8] in the affirmative and was later extended to the local version in [9] and the multisublinear version in [5,10].…”