Surjectivity of Mean Value Operators on Noncompact Symmetric Spaces

TL;DR

This paper investigates the surjectivity of mean value operators on noncompact symmetric spaces, establishing conditions under which these operators are surjective on smooth functions, especially focusing on complex, rank one, and higher rank cases.

Contribution

It proves surjectivity of mean value operators on certain noncompact symmetric spaces, extending known results to higher rank spaces within specific Weyl subchambers.

Findings

Surjectivity holds for complex and rank one symmetric spaces.

For higher rank spaces, surjectivity is valid within an appropriate Weyl subchamber.

The results generalize previous understanding of mean value operators on symmetric spaces.

Abstract

Let $X=G/K$ be a symmetric space of the non-compact type. We prove that the mean value operator over translated $K$-orbits of a fixed point is surjective on the space of smooth functions on $X$ if $X$ is either complex or of rank one. For higher rank spaces it is shown that the same statement is true for points in an appropriate Weyl subchamber.