A remarkable class of second order differential operators on the Heisenberg group ${\mathbb H}_2$

Abstract

Let L be a differential operator on \({\mathbb R}^5\) whose principal part is of the form \(\!\sum_{\!j,k=1}^2\! \alpha_{jk}\! X_j Y_k\), where \(X_j=\partial_{x_j}\) and \(Y_j=\partial_{y_j}+x_j\partial_u , j=1,2\), are the usual vector fields generating the Lie algebra of the Heisenberg group \({\mathbb H}_2\). We study the problem of local solvability of these doubly characteristic operators. The whole class of operators splits into three subclasses, depending on the sign of a respective determinant. The operators in the first subclass, when the determinant is negative, are generically non-solvable. The operators in the second subclass, when the determinant is positive, are solvable, for arbitrary left-invariant lower order terms, provided that the coefficient matrix \((\alpha_{j,k})\) is non-degenerate. This fact seems remarkable, since many of these operators have the property that the values taken by their principal symbol are not contained in any proper subcone of the complex plane. Under suitable conditions, solvability even holds in the presence of non-invariant lower order terms.