Homogeneous even kernels on surfaces
Let \(K(x)= {P_ k(x)\over | x|^{k+n}}\), where \(k\) is a positive integer, \(P_ k\) is a homogeneous polynomial of degree \(k\) defined on \(\mathbb{R}^{n+1}\). It was known previously that if \(P_ k\) is odd, then for a wide class of \(n\)-dimensional surfaces the principal value singular integral introduced by the kernel \(K\) defined on some nice classes of functions on these surfaces can be extended to \(L^ 2\)- bounded operators. In this paper we proved that for an even kernel of the given type, if the associated maximal integrals of the indicator functions on a set of \(n\)-dimensional surfaces are a.e. finite, and, if the set of surfaces has a set of tangent hyperplanes of a certain variety of directions, then the kernel must be identically zero.
scientific article; zbMATH DE number 704911