Fatou-type theorems for general approximate identities

Abstract

For functions $f \in L^{1}(\mathsf{R}^n)$ we consider extensions to $\mathsf{R}^n \times \mathsf{R}^{+}$ given by convolving $f$ with an approximate identity. For a large class of approximate identities we obtain a Fatou-type theorem where the convergence regions are sometimes effectively larger than the non-tangential ones. We then study a more restricted class of approximate identities for which the convergence regions are shown to be optimal. Finally we will consider products of approximate identities. The results extend previous results by Sjöogren, Rönning and Brundin .