A family of Riesz distributions for differential forms on Euclidian space

TL;DR

This paper introduces a new family of operator-valued distributions on Euclidean space that generalize Riesz distributions for functions, exploring their properties and connections to conformal geometry and group representations.

Contribution

It extends Riesz distributions to differential forms, providing a framework for analyzing their properties and applications in conformal geometry and representation theory.

Findings

Developed a family of distributions acting on differential forms

Analyzed meromorphic continuation and residues of these distributions

Connected distributions to conformal geometry and group representations

Abstract

In this paper we introduce a new family of operator-valued distributions on Euclidian space acting by convolution on differential forms. It provides a natural generalization of the important Riesz distributions acting on functions, where the corresponding operators are $(-\Delta)^{-\alpha/2}$, and we develop basic analogous properties with respect to meromorphic continuation, residues, Fourier transforms, and relations to conformal geometry and representations of the conformal group.