  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.
