Weighted estimates of singular integrals and their applications

We give a survey of research on the problem of single-weighted and double-weighted estimates of strong and weak types for the Hardy-Littlewood maximal function, Riesz potentials, singular integral operators, and harmonic functions. Necessary and sufficient conditions on the weight are given under which weighted estimates are valid (Muckenhoupt's A_p-condition, Sawyer's condition, etc.). Special attention is given to papers which appeared after 1980 and the latest results, published as reports and preprints.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 21, pp. 42–129, 1983.