On the convergence of singular integrals of vector-valued functions

Abstract

The pointwise convergence of abstract singular convolution integrals K(•,v) * f (x - •) depending on two parameters x,v (x ϵ R, v belongs to a metric space E) is studied. The convergence of the parameters is restricted to some subsets o R x E (i.e. the Fatou’s type of the convergence is discussed). Parameter x tends to a generalized D or L-point of the Banach space-valued function f, whereas v tends to an accumulation point of E. Some applications of the obtained results to the norm-convergence of real-valued singular integrals and partial differential equations are presented.