Compactness and Complete Continuity for Commutators of Rough Singular Integrals and Their Maximal Operators

Let TΩ denote the Calderón–Zygmund singular integral operator with a kernel Ω on Sn−1 that lacks regularity and merely satisfies the size condition Ω∈L(logL)2(Sn−1). Let TΩ,b be the commutator of TΩ with b∈CMO(Rn). In this paper, we prove that the discrete maximal operator TΩ,b∗∗ associated with TΩ,b is completely continuous on the weighted space Lp(Rn,|x|γ) for all 1<p<∞ and −1<γ<p−1. Furthermore, we establish the compactness and complete continuity of both TΩ,b and TΩ,b∗∗ on the Morrey space Lp,λ(Rn) for Ω∈L(logL)2(Sn−1) or Ω∈Lq(Sn−1)(q>1). The maximal commutator TΩ,b∗ is also discussed.