Bulletin of the
Korean Mathematical Society BKMS

In the present paper we establish the boundedness and continuity of the higher order maximal commutators with Lipschitz symbols on the Sobolev spaces, Triebel–Lizorkin spaces and Besov spaces. More precisely, let 0 ≤ α < d and Mkb,α(k ≥ 1) be the k-th order fractional maximal commutator. When α = 0, we denote Mkb,α = Mkb. We prove that Mkb,α maps the first order Sobolev spaces W1,p(Rd) boundedly and continuously to W1,q(Rd) for 1 < p < q < ∞ and 1/q = 1/p−α/d if b belongs to the inhomogeneous Lipschitz space Lip(Rd). We also show that if 0 < γ ≤ 1,0 < s < γ, 1 < p,q < ∞ and b ∈ Lipγ(Rd), then Mkbis bounded and continuous from the fractionalSobolev spaces Ws,p(Rd) to itself, from the inhomogeneous Triebel–Lizorkin spaces Fp,qs (Rd) to itself and from the inhomogeneous Besov spaces Bp,qs (Rd) to itself.

Keywords: Higher order maximal commutators, Boundedness and continuity, Lipschitz spaces, Sobolev spaces, Triebel-Lizorkin and Besov spaces

MSC numbers: 2020 Mathematics Subject Classification. Primary 42B25, 46E3