Non-homogeneous harmonic analysis: 16 years of development

This survey contains results and methods in the theory of singular integrals, a theory which has been developing dramatically in the last 15–20 years. The central (although not the only) topic of the paper is the connection between the analytic properties of integrals and operators with Calderón–Zygmund kernels and the geometric properties of the measures. The history is traced of the classical Painlevé problem of describing removable singularities of bounded analytic functions, which has provided a strong incentive for the development of this branch of harmonic analysis. The progress of recent decades has largely been based on the creation of an apparatus for dealing with non-homogeneous measures, and much attention is devoted to this apparatus here. Several open questions are stated, first and foremost in the multidimensional case, where the method of curvature of a measure is not available.
Bibliography: 128 titles.