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

Abstract

Let M be the Hardy–Littlewood maximal function and let [b,M] be its corresponding commutator. For 1<p1,p2,p,q<∞ and 1/p=1/p1+1/p2, we show that the commutator [b,M] is bounded and continuous from Sobolev space Ws,p1true(Rdtrue) to Sobolev space Ws,p(double-struckRd) for 0≤s≤1 when b∈Ws,p2true(Rdtrue), from Triebel–Lizorkin space Fsp1,q(double-struckRd) to Fsp,q(double-struckRd) if b∈Fsp2,q(double-struckRd) and from Besov space Bsp1,q(double-struckRd) to Bsp,q(double-struckRd) if b∈Bsp2,q(double-struckRd) and 0<s<1.

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

“…For other interesting works related to this theory, we suggest the readers to consult [19][20][21][22], among others. Very recently, Liu et al [23] investigated the regularity of commutators of the Hardy-Littlewood maximal function. Precisely, let b be a locally integrable function defined on ℝ n , we define the commutator of the Hardy-Littlewood maximal function ½b, M by…”