This paper is written according to author’s series of talks presented at the Spring School: Nonlinear Analysis, Function Spaces and Applications 9. It explains some results of Calderón-Zygmund theory (Calderón-Zygmund decomposition, the weak \((1,1)\) boundedness of Calderón-Zygmund operators, Cotlar’s inequality, the T(1) theorem, and the definitions of BMO and Hardy spaces) with underlying measure in \(\mathbb{R}^d\), \(d\in\mathbb{N}\), which need non be doubling. It is known as non homogeneous Calderón-Zygmund theory. Moreover, the author describes the relationship between the Cauchy transform and Menger curvature and shows its applications to the study of analytic capacity and the so called Painlevéproblem.
Let us remark that some parts of this paper follow quite closely previous author’s surveys such as [in: Laptev, Ari (ed.), Proceedings of the 4th European congress of mathematics (ECM), Stockholm, Sweden, June 27–July 2 (2004), Zürich: EMS, 459–476 (2005; Zbl 1095.30021)] and [in: Seminar of mathematical analysis. Proceedings of the lecture notes of the seminar, Universities of Malaga and Seville, Spain, September 2003–June 2004, Colección Abierta 71, 239–271 (2004; Zbl 1079.42007)]. However, the present paper contains more information and details, such as a somewhat new proof of the T(1) theorem for the Cauchy transform.
For the entire collection see [Zbl 1262.35002].