Weakly Singular Integral Operators in Weighted L∞–Spaces

Abstract

We study integral operators on (−1, 1) with kernels k(x, t) which may have weak singularities in (x, t) with x ∈N1, t ∈N2, or x=t, where N1,N2 are sets of measure zero. It is shown that such operators map weighted L∞–spaces into certain weighted spaces of smooth functions, where the degree of smoothness is the higher the smoother the kernel k(x, t) as a function in x is. The spaces of smooth function are generalizations of the Ditzian-Totik spaces which are defined in terms of the errors of best weighted uniform approximation by algebraic polynomials.