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

Let M be the Hardy–Littlewood maximal function and let [ b , M ] be its corresponding commutator. For 1 < p 1 , p 2 , p , q < ∞ and 1 / p = 1 / p 1 + 1 / p 2 , we show that the commutator [ b , M ] is bounded and continuous from Sobolev spaceW s , p 1( R d )to Sobolev spaceW s , p ( R d ) for 0 ≤ s ≤ 1 when b ∈ W s , p 2( R d ) , from Triebel–Lizorkin spaceF s p 1 , q ( R d ) toF s p , q ( R d ) if b ∈ F s p 2 , q ( R d ) and from Besov spaceB s p 1 , q ( R d ) toB s p , q ( R d ) if b ∈ B s p 2 , q ( R d ) and 0 < s < 1 .