Analysis Seminar on "Boundedness and compactness of commutator of Cauchy integrals for domains in C^n with minimal smoothness" by Ji Li (Macquarie University, Australia)
Title: Boundedness and compactness of commutator of Cauchy integrals for domains in C^n with minimal smoothness
Abstract: It is well-known that the commutator of Riesz transform (Hilbert transform in dimension 1) and a symbol b is bounded on L^2(R^n) if and only if b is in the space of bounded mean oscillation BMO(R^n) (Coifman--Rochberg--Weiss). Moreover, the commutator is compact on L^2(R^n) if and only if b is in the space of vanishing mean oscillation VMO(R^n) (Uchiyama).
We provide such characterisations of the commutator of Cauchy type integrals for domains in C^n with minimal smoothness (studied by Lanzani--Stein), which are singular integrals with non-smooth kernels (beyond the standard frame of Calder\'on--Zygmund operators). It is worth to point out that our method for compactness is new and can recover the known results for standard Calder\'on--Zygmund operators.
The results we provide here are based on recent joint works with Xuan Thinh Duong, Michael Lacey and Brett D. Wick.
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