The Cauchy integral, analytic capacity, and uniform rectifiability.

*(English)*Zbl 0897.42007Let \(E\) 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.

Reviewer: S.G.Staples (Fort Worth)

##### 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 |