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The Cauchy integral, analytic capacity, and uniform rectifiability. (English) Zbl 0897.42007
Let $$E$$ be a closed subset of the complex plane. If there exists some constant $$0<M<\infty$$ such that $$r/M \leq \mathcal H^{1} (E\cap \Delta (z,r))\leq rM$$ for all $$z\in E$$ and $$0<r<\text{diam} E$$, then we say $$E$$ is Ahlfors-David or (AD)-regular. Here $$\mathcal H^{1}$$ is the 1-dimensional Hausdorff measure and $$\Delta (z,r)$$ the closed disc with center $$z$$ and radius $$r$$. The authors prove that if $$E$$ is (AD)-regular then (a) the Cauchy integral operator, $$C_{E}$$, is bounded in $$L^{2}(E)$$ if and only if $$E$$ is contained in a AD-regular curve and (b) the analytic capacity of $$E$$ is zero if and only if $$\mathcal H ^{1} (E\cap \Gamma) = 0$$ for every rectifiable curve $$\Gamma$$. In the case that $$E$$ is a rectifiable curve, Theorem (a) was proved by G. David [Ann. Sci. Éc. Norm. Supér., IV. Sér. 17, 157-189 (1984; Zbl 0537.42016)]; it follows immediately that if $$E$$ is contained in an AD-regular curve then $$C_{E}$$ is bounded in $$L^{2}(E)$$. The proof of the converse is the heart of the current paper. Once (a) has been established, the authors can utilize a result of M. Christ [Regional Conference Series in Mathematics 77. Providence, RI: American Mathematical Society, 132 p. (1990; Zbl 0745.42008); Colloq. Math. 60/61, No. 2, 601-628 (1990; Zbl 0758.42009)] which relates the $$L^{2}$$ boundedness of $$C_{E}$$ and the analytic capacity of $$E$$ for AD-regular sets.

MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 30C85 Capacity and harmonic measure in the complex plane 30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane 31A10 Integral representations, integral operators, integral equations methods in two dimensions
MathOverflow Questions:
Origin of term Ahlfors-David regular
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