Local solvability for positive combinations of generalized sub-Laplacians on the Heisenberg group

Abstract:

As one step in a program to understand local solvability of complex coefficient second order differential operators on the Heisenberg group in a complete way, solvability of operators of the form $\Delta _{S,\alpha }=\Delta _S +i\alpha U$, where the leading term $\Delta _S$ is a “positive combination of generalized and degenerate generalized sub-Laplacians”, has been studied in a recent article by M. Peloso, F. Ricci and the first-named author (J. Reine Angew Math. 513 (1999)). It was shown that there exists a discrete set of “critical” values $E\subset \mathbb {C}$, such that solvability holds for $\alpha \not \in E$. The case $\alpha \in E$ remained open, and it is the purpose of this note to close this gap. Our results extend corresponding results in another article by the above-mentioned authors (J. Funct. Anal. 148 (1997)), by means of an even simplified approach which should allow for further generalizations.