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

We prove Fatou type theorem on almost everywhere convergence of convolution integrals in spaces $L^p\,(1<p<\infty)$ for general kernels, forming an approximate identity. For a wide class of kernels we show that obtained convergence regions are optimal in some sense. It is also established a weak boundedness of the corresponding maximal operator in $L^p\,(1\le p<\infty)$.