Commutators of Cauchy-Szego type integrals for domains in C^n with minimal smoothness

Abstract

Let D ⊂ C n be a bounded, strongly pseudoconvex domain whose boundary bD satisfies the minimal regularity condition of class C 2 . We characterize boundedness and compactness in L p (bD, ω), for 1 < p < ∞, of the commutator [b, Sω] where Sω is the Cauchy-Szegő (orthogonal) projection of L 2 (bD, ω) onto the holomorphic Hardy space H 2 (bD, ω) and the measure ω belongs to a family (the "Leray Levi-like" measures) that includes induced Lebesgue measure σ. We next consider a much larger family of measures {Ω} modeled after the Muckenhoupt weights for σ: we define the holomorphic Hardy spaces H p (bD, Ω) for any Ap-like measure Ω and we characterize boundedness and compactness of [b, SΩ] in L 2 (bD, Ω) for any A2-like measure Ω. Earlier closely related results rely upon an asymptotic expansion, and subsequent pointwise estimates, of the Cauchy-Szegő kernel that are not available in the settings of minimal regularity of bD and/or Ap-like measures.Among the new main tools are • operator identities for [b, Sω] and for [b, SΩ] inspired by the classical Kerzman-Stein equation for Sσ in L 2 (bD, σ) [22] and its 2017 variant in L p (bD, ω) [32], where p = 2; • sharp weighted estimates in L p (bD, Ω) for a family of Cauchy type integral operators {Cǫ}ǫ (related to SΩ via the Kerzman-Stein equations) and for their commutators {[b, Cǫ]}ǫ; • cancellation estimates in L p (bD, Ω) for the symmetrized truncation of the aforementioned {Cǫ}ǫ with quantitative norm bounds displaying explicit dependence on ǫ.

Commutators of maximal functions on spaces of homogeneous type and their weighted, local versions

“…We remark that a concrete example of the space of homogeneous type (X, d, µ) with µ(X) < ∞ and diam(X) < ∞ is the boundary of a bounded strictly pseudoconvex domain in C n , see for example the recent works in [14], [30] and [32]. To be more precise, we recall the bounded domain D from [32] with defining function ρ, which means that D = {z ∈ C n : ρ(z) < 0} with ρ : C n → R and boundary bD.…”

Boundedness and Compactness of Cauchy-Type Integral Commutator on Weighted Morrey Spaces

“…It is important to study the harmonic analysis of such Cauchy-Szegő operators and Cauchy integral operators in complex analysis, for example, to establish the theory of holomorphic Hardy spaces over such domains. Duong et al [4] proved the characterization of boundedness and compactness for commutators of such Cauchy type integral operators. In an abuse of notation, we omit the subscript .…”

“…THEOREM A [4]. Suppose D ⊂ C n , n ≥ 2, is a bounded domain whose boundary is of class C 2 and is strongly pseudoconvex.…”

“…The main result of our paper is to characterize the boundedness and compactness of the commutator of Cauchy type integral C on the weighted Morrey space, following the ideas and approach in [4].…”

“…They obtained the L p (bD) boundedness (1 < p < ∞) of the Cauchy-Leray transform C by showing that the kernel K(w, z) of C satisfies the standard size and smoothness conditions of Calderón-Zygmund operators (for details of these definitions and notation we refer the reader to Section 3), and that C satisfies a suitable version of the T(1) theorem. In [4], the authors also obtained the characterization of boundedness and compactness for commutators of such transforms. Following a similar approach to that in the proof of Theorems 1.1 and 1.2, we obtain the following results on the Cauchy-Leray transform and its commutator.…”