Bilipschitz maps , analytic capacity , and the Cauchy integral
نویسنده
چکیده
Let φ : C → C be a bilipschitz map. We prove that if E ⊂ C is compact, and γ(E), α(E) stand for its analytic and continuous analytic capacity respectively, then C−1γ(E) ≤ γ(φ(E)) ≤ Cγ(E) and C−1α(E) ≤ α(φ(E)) ≤ Cα(E), where C depends only on the bilipschitz constant of φ. Further, we show that if μ is a Radon measure on C and the Cauchy transform is bounded on L2(μ), then the Cauchy transform is also bounded on L(φ♯μ), where φ♯μ is the image measure of μ by φ. To obtain these results, we estimate the curvature of φ♯μ by means of a corona type decomposition.
منابع مشابه
Analytic Capacity , and the Cauchy Integral
Let φ : C → C be a bilipschitz map. We prove that if E ⊂ C is compact, and γ(E), α(E) stand for its analytic and continuous analytic capacity respectively, then Cγ(E) ≤ γ(φ(E)) ≤ Cγ(E) and Cα(E) ≤ α(φ(E)) ≤ Cα(E), where C depends only on the bilipschitz constant of φ. Further, we show that if μ is a Radon measure on C and the Cauchy transform is bounded on L(μ), then the Cauchy transform is als...
متن کاملAnalytic capacity, rectifiability, and the Cauchy integral
A compact set E ⊂ C is said to be removable for bounded analytic functions if for any open set containing E, every bounded function analytic on \ E has an analytic extension to . Analytic capacity is a notion that, in a sense, measures the size of a set as a non removable singularity. In particular, a compact set is removable if and only if its analytic capacity vanishes. The so-called Painlevé...
متن کامل$L_{p;r} $ spaces: Cauchy Singular Integral, Hardy Classes and Riemann-Hilbert Problem in this Framework
In the present work the space $L_{p;r} $ which is continuously embedded into $L_{p} $ is introduced. The corresponding Hardy spaces of analytic functions are defined as well. Some properties of the functions from these spaces are studied. The analogs of some results in the classical theory of Hardy spaces are proved for the new spaces. It is shown that the Cauchy singular integral operator is...
متن کاملEstimate of the Cauchy Integral over Ahlfors Regular Curves
We obtain the complete characterization of those domains G ⊂ C which admit the so called estimate of the Cauchy integral, that is to say, ̨̨̨R ∂G f(z) dz ̨̨̨ ≤ C(G) ‖f‖∞ γ(E) for all E ⊂ G and f ∈ H∞(G \ E), where γ(E) is the analytic capacity of E. The corresponding result for continuous functions f and the continuous analytic capacity α(E) is also proved.
متن کاملOn Analytic Capacity of Portions of Continuum and Atheorem of Guy Davidf
We are going to \compute" in metric terms the analytic capacity of the intersection of a continuum and a half-plane (or a disc, or any domain with piecewise smooth boundary). We use this calculation to provide a purely analytic proof to a theorem of Guy David characterizing the rectiiable curves on the plane for which the Cauchy integral operator is bounded on L 2 (ds). In doing that we use als...
متن کامل