Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

Abstract

Let CΓ be the Cauchy integral operator on a Lipschitz curve Γ. In this article, the authors show that the commutator [b,CΓ] is bounded (resp, compact) on the Morrey space Lp,λfalse(double-struckRfalse) for any (or some) p ∈ (1,∞) and λ ∈ (0,1) if and only if b∈0.1emBMOfalse(double-struckRfalse) (resp, CMOfalse(double-struckRfalse)). As an application, a factorization of the classical Hardy space H1false(double-struckRfalse) in terms of CΓ and its adjoint operator is obtained.

Beurling–Ahlfors Commutators on Weighted Morrey Spaces and Applications to Beltrami Equations

“…In this section, we first obtain a simple but useful auxiliary lemma (see Lemma 2.1 below), which is on the domination of |b(z) − α Q (b)| for a given real-valued function b ∈ L 1 loc (C) by the difference |b(z) − b(u)| pointwise on subsets of Q × Q, where Q and Q are squares and α Q (b) is the median value of b over Q . Compared to [22,27], our method adopted in the proof of Theorem 1.3 avoids the use of the so-called local mean oscillation. Section 3 is devoted to the proof of Theorem 1.4 and is divided into two subsections.…”

“…Thus, in the proof of Theorem 1.4(i), we use some ideas from [21,7] Subsection 3.2 is devoted to the proof of Theorem 1.4(ii). As in the unweighted case (see, for example, [28,27]), we first obtain a lemma for the upper and the lower bounds of integrals of [b, B] f j related to certain squares Q j , for any real-valued function b ∈ BMO(C) and proper functions f j defined by Q j with j ∈ N; see Lemma 3.5 below. Since a general A p (C) weight is not invariant under translations, besides Lemma 3.5, we also obtain a variant of Lemma 3.5, where the geometrical relation of {Q j } j∈N are involved; see Lemma 3.6 below.…”

“…Since Theorem 1.3(i) is a corollary of [20,Theorem 3.4], it suffices to prove Theorem 1.3(ii). Compared to the method used in [22,27], our method avoids the use of the so-called local mean oscillation; see also [12,15].…”

“…Lemma 3.5 is sufficient to derive the necessity of the compactness of Calderón-Zygmund commutators in unweighted case; see, for example, [27]. For weighted case, since a general weight w ∈ A p (C) is not invariant under translations, we also need the following Lemma 3.6 to deal with some tricky situations.…”

Compactness of Riesz transform commutator on stratified Lie groups

“…Uchiyama [25] then showed that [b, R j ] is compact on L p (R n ) with 1 < p < ∞ if and only if b ∈ VMO(R n ), the space of functions with vanishing mean oscillation on R n . Later on and recently, there has been an intensive study of the compactness of commutators of singular integrals in many different settings, such as the Riesz transform associated with Bessel operator on the positive real line, the Cauchy's integrals on the real line, the Calderón-Zygmund operator associated with homogeneous kernels Ω(x) |x| n on R n , and the multilinear Riesz transforms, see for example [4,9,13,17,18,19,23] and related references therein.…”