Weighted spherical means generated by generalized translation and general Euler-Poisson-Darboux equation

TL;DR

This paper studies weighted spherical means generated by generalized translations and their relation to a generalized Euler-Poisson-Darboux equation, highlighting properties of solutions including the Asgeirsson property in a complex differential operator context.

Contribution

It introduces a new framework connecting weighted spherical means with a generalized Euler-Poisson-Darboux equation and establishes the Asgeirsson property for solutions involving singular Bessel operators.

Findings

Established the Asgeirsson property for solutions of the ultrahyperbolic equation.

Connected weighted spherical means with generalized translation and Euler-Poisson-Darboux equations.

Extended classical results to include singular differential Bessel operators.

Abstract

We consider the spherical mean generated by a multidimensional generalized translation and general Euler-Poisson-Darboux equation corresponding to this mean. The Asgeirsson property of solutions of the ultrahyperbolic equation that includes singular differential Bessel operators acting by each variable is provided.