On generalizations of Fatou's theorem for the integrals with general kernels

Abstract

We define $\lambda (r)$-convergence, which is a generalization of nontangential convergence in the unit disc. We prove Fatou-type theorems on almost everywhere nontangential convergence of Poisson-Stiltjes integrals for general kernels $\{{\phi }_r\}$, forming an approximation of identity. We prove that the bound \md0 \limsup_{r\to 1}λ(r) \|φ_r\|_\infty<\infty \emd is necessary and sufficient for almost everywhere $\lambda (r)$-convergence of the integrals \md0 \int_\ZT φ_r(t-x)dμ(t). \emd

14 pages

Cited by in corpus (1)

• On Generalizations of Fatou's Theorem in $L^p$ for Convolution Integrals with General Kernels