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The theory of singular integral operators was developed by Calderón and Zygmund in the 1950s, and since then it has become a central topic of study in harmonic analysis, with many applications in both pure and applied mathematics. Two relevant examples of such operators are the Cauchy and Riesz transforms, which play a fundamental role in complex and harmonic analysis, and which have plenty of applications in PDE's, geometric measure theory, and mathematical physics. In this book, some topics concerning the Cauchy and Riesz transforms and other singular integrals are studied from the geometric analysis point of view. Most of these topics are connected to interesting open problems, such as the relation between Riesz transforms and rectifiability. Geometric properties and applications of some capacities defined in terms of the Cauchy and Riesz transforms are also studied. This book, which is the publication of the author's PhD dissertation, is suitable for graduate students and researchers in mathematics interested in Calderón-Zygmund theory and geometric analysis.
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PhD in Mathematics at Universitat Autònoma de Barcelona (Spain), 2011. Postdoctoral Research Position at Universidad del País Vasco (Spain). Research interests: harmonic analysis (in particular Calderón-Zygmund theory) and geometric measure theory. Partial differential equations and rational approximation.
