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 $\{\varphi_r\}$, forming an approximation of identity. We prove that the bound \md0 \limsup_{r\to 1}\lambda(r) \|\varphi_r\|_\infty<\infty \emd is necessary and sufficient for almost everywhere $\lambda(r)$-convergence of the integrals \md0 \int_\ZT \varphi_r(t-x)d\mu(t). \emd