On the symbol calculus for multidimensional Hausdorff operators

TL;DR

This paper develops a symbol calculus for multidimensional Hausdorff operators on L^2(R^n), considering cases with commuting self-adjoint matrices and positive definite matrices, using diagonalization techniques.

Contribution

It introduces a new symbol calculus framework for multidimensional Hausdorff operators, extending previous diagonalization methods to broader matrix classes.

Findings

Established symbol calculus for commuting self-adjoint matrices.

Extended calculus to positive definite matrices.

Utilized diagonalization of normal Hausdorff operators.

Abstract

The aim of this work is to derive a symbol calculus on $L^2(\mathbb{R}^n)$ for multidimensional Hausdorff operators. Two aspects of this activity result in two almost independent parts. While throughout the perturbation matrices are supposed to be self-adjoint and form a commuting family, in the second part they are additionally assumed to be positive definite. What relates these two parts is the powerful method of diagonalization of a normal Hausdorff operator elaborated earlier by the second author.