Singular integrals and rectifiability

Abstract

The problems addressed in this dissertation live in the intersection between Harmonic Analysis and Geometric Measure Theory, and so one should say that they belong to the area of Geometric Analysis. Precisely, we have analyzed relationships between singular integral operators such as the Riesz transform with respect to general Borel measures in the Euclidean space, and metric or geometric properties of those measures or their supports. In the next few pages we summarize the workflow we have followed in the development of this dissertation and the results we have obtained, as well as some definitions of the concepts that are needed to understand these results. The rest of pertinent definitions, auxiliary results and proofs will be found in the next chapters. We wish to remark, as well, that the results in Chapter 1 can be found at [G1], the ones in Chapter 2 can be found at [G2] and the ones in Chapter 3 can be found at [GT], which is a collaboration with Tolsa. This does not mean that Chapter 1 and Chapter 2 have been developed independently by the author of this dissertation, as all the work presented here has been done under the guidance of Professor Tolsa