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

## TL;DR

This paper extends Fatou's theorem to $L^p$ spaces for convolution integrals with general kernels, establishing convergence and boundedness results that generalize classical harmonic analysis theorems.

## Contribution

It introduces a generalized Fatou theorem for convolution integrals with broad kernel classes in $L^p$ spaces, including optimal convergence regions and weak boundedness of maximal operators.

## Key findings

- Proved almost everywhere convergence of convolution integrals in $L^p$ with general kernels.
- Identified optimal convergence regions for a wide class of kernels.
- Established weak boundedness of the maximal operator in $L^p$.

## Abstract

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