Operators from Classical Harmonic Analysis

Abstract

Two of the most important operators arising in harmonic analysis are the Fourier transform and convolutions (which include translation operators via convolution with Dirac point measures). The aim of this final chapter is to make a detailed analysis of these two classes of operators, acting in L^p-spaces, from the viewpoint of their optimal domain and properties of the corresponding extended operator. In particular, for the well-known class of L^q-improving measures, it turns out that the corresponding convolution operators can be characterized as precisely those which are p-th power factorable for a suitable range of p; see Section 7.5. This makes a close and important connection between the results of Chapter 5 and the classical family of convolution operators.