Solvability of dissipative second order left-invariant differential operators on the Heisenberg group

Abstract

We prove local solvability for large classes of operators of the form \[ L=\sum_{j,k=1}^{2n}a_{jk}V_jV_k+i\alpha U,\] where the \(V_j\) are left-invariant vector fields on the Heisenberg group satisfying the commutation relations \([V_j,V_{j+n}]=U\) for \(1\le j\le n\), and where \(A=(a_{jk})\) is a complex symmetric matrix with semi-definite real part. Our results widely extend all of the results for the case of non-real, semi-definite matrices \(A\) known to date, in particular those obtained recently jointly with F. Ricci under Sj\"ostrand's cone condition.