Remarks on Almost Everywhere Convergence and Approximate Identities

Abstract

We prove almost everywhere convergence for convolutions of locally integrable functions with shrinking L¹ dilations of a fixed integrable kernel with an integrable radially decreasing majorant. The set on which the convergence holds is an explicit subset of the Lebesgue set of the locally integrable function of full measure. This result can be viewed as an extension of the Lebesgue differentiation theorem in which the characteristic function of the unit ball is replaced by a more general kernel. We obtain a similar result for multilinear convolutions.