On partially hypoelliptic operators. Part I: Differential operators

TL;DR

This paper investigates variable coefficient, self-adjoint, partially hypoelliptic differential operators, providing fundamental solutions and spectral kernel estimates, with generalizations to pseudo differential operators discussed in subsequent work.

Contribution

It offers a detailed analysis and construction of fundamental solutions for partially hypoelliptic differential operators with variable coefficients, extending previous results.

Findings

Constructed fundamental solutions in a suitable topology

Provided estimates for spectral kernels

Extended analysis to variable coefficient operators

Abstract

This article gives a fundamental discussion on variable coefficients, self-adjoint, formally partially hypoelliptic differential operators. A generalization of the results to pseudo differential operators, is given in a following article in ArXiv. Close to N. Nilsson "Some estimates for spectral functions connected with formally hypoelliptic differential operators" in Arkiv f\"or matematik ,10, 1972, we give a construction and estimates for a fundamental solution to the operator in a suitable topology. We further give estimates of the corresponding spectral kernel.