SPHERICAL HARMONICS AND HARDY'S INEQUALITIES (Mathematical aspects of quantum fields and related topics)

Abstract

We consider the derivative operators for radial direction and spherical direction. We also investigate the operator which takes the spherical average for functions. We reconfirm those properties with particular attention to orthogonality. As an application, the Hardy type inequality is presented with spherical derivatives in the framework of equalities. This clarifies the difference between contribution by radial and spherical derivatives in the improved Hardy inequality as well as nonexistence of nontrivial extremizers without compactness arguments.