Randomization of singular integrals: the way to survive in bad neighborhoods
 

Singular integrals in very bad behaving measure spaces are coming into the focus of attention because of problems of Geometric Measure Theory (GMT), where a priori measures have no smoothness. Another source is the sharp weighted estimates of singular operators often appearing from PDE. The sharpness makes necessary the decoupling of measure and kernel, sometimes a quite radical one. The typical setting would be a metric space of homogeneous type (so it has a doubling measure), but the singular operator on it is considered with respect to another non-doubling and very badly behaving measure. Examples, where this happens (or may happen), are Painlevé, Denjoy, Vitushkin's problems; David–Semmes problem; analysis on the boundaries of pseudo-convex domains that goes beyond the scope of Carnot–Carathéodory spaces; two weight Hilbert transform; one weight sharp estimates of Calderón–Zygmund operators, the problems of Geometric Measure Theory, the problems of perturbation of normal operators, etcetera? We will show that probabilistic methods of treatment of such operators in "bad neighborhoods" is usually a powerful remedy.