Commutators of singular integrals revisited

Abstract

We obtain a Bloom‐type characterization of the two‐weighted boundedness of iterated commutators of singular integrals. The necessity is established for a rather wide class of operators, providing a new result even in the unweighted setting for the first order commutators.

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

“…These results were further extended to the weighted Lebesgue space L p w (C) with p ∈ (1, ∞) and w ∈ A p (C) by Clop and Cruz [7], where they also obtained a priori estimate in L p w (C) for the generalized Beltrami equation and the regularity for the Jacobian of certain quasiconformal mappings. For more results on the boundedness and the compactness of Calderón-Zygmund commutators on function spaces and their applications, please see [15,8,21,19,23,22] and references therein.…”

“…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.…”

Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

“…Inspired by the recent work, 35 we show Theorem 1.4 by means of the so-called local mean oscillations of functions. Recall that, given a measurable function f on R and an interval I ⊂ R, the local mean oscillation (f; I) of f on I is defined by setting…”

“…Inspired by the recent work, we show Theorem by means of the so‐called local mean oscillations of functions. Recall that, given a measurable function f on and an interval , the local mean oscillation ω μ ( f ; I ) of f on I is defined by setting where μ ∈ (0,1) and f ∗ denotes the nonincreasing rearrangement of f , namely, for any t ∈ [0, ∞ ), …”

“…We first give the proof of Theorem in Section and then the proof of Theorem in Section . In the proof of Theorem , inspired by Lerner et al, we obtain an auxiliary result suitable for C Γ (see Lemma below), which is on the domination of the local mean oscillation of b on a given interval I by the difference of | b ( x ) − b ( y )| pointwise on subsets of I × 5 I , where for any given interval I : = I ( x , r ) with and r ∈ (0, ∞ ), 5 I : = I ( x ,5 r ). Compared with the argument used in the proof of Lerner et al , proposition 3.1 therein, the argument used here is simpler due to the specific structure of the kernel C Γ .…”