H\"ormander type pseudodifferential calculus on homogeneous groups

TL;DR

This paper develops a H"ormander pseudodifferential calculus on general homogeneous groups, extending classical Euclidean results to a broader setting without restricting to classical symbols or specific groups.

Contribution

It introduces a new framework for pseudodifferential operators on homogeneous groups, characterizing multipliers and convolution operators of any order, and establishing their algebraic properties.

Findings

Characterization of inverse Fourier transforms of multipliers of any order

Composition of convolution operators of orders j_1 and j_2 results in an operator of order j_1 + j_2

Extension of H"ormander calculus beyond classical symbols and specific groups

Abstract

We produce, on general homogeneous groups, an analogue of the usual H\"ormander pseudodifferential calculus on Euclidean space, at least as far as products and adjoints are concerned. In contrast to earlier works, we do not limit ourselves to analogues of classical symbols, nor to the Heisenberg group. The key technique is to understand ``multipliers'' of any given order j, and the operators of convolution with their inverse Fourier transforms, which we here call convolution operators of order j. (Here a ``multiplier'' is an analogue of a H\"ormander-type symbol a(x,\xi), which is independent of x.) Specifically, we characterize the space of inverse Fourier transforms of multipliers of any order j, and use this characterization to show that the composition of convolution operators of order j_1 and j_2 is a convolution operator of order j_1+j_2.