Calderón–Zygmund theory for kernels with nondiscrete sets of singularities

An approach encompassing diverse locations of the singularities of convolution kernels by a single method is presented. In particular, the author introduces the notion of a so-called supersingular kernel, whose singularities lie on a set of arbitrary structure, in general, and a theorem on the continuity in $L^p(\mathbb R^N)$, $1<p<\infty$, of the operator of convolution with it is established. Together with the theorem on convergence almost everywhere of the sequence of convolutions defining this operator, with cutoffs of the kernel defined in a special way, it is a generalization of fundamental results of Calderón–Zygmund.