  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 $L^p\,(1\le p<\infty)$.

Classical Analysis and ODEs