Non-solvability for a class of left-invariant second-order differential operators on the Heisenberg group

Abstract

We study the question of local solvability for second-order, left-invariant differential operators on the Heisenberg group H n \mathbb {H}_n , of the form \[ P Λ = ∑ i , j = 1 n λ i j X i Y j = t X Λ Y , \mathcal {P}_\Lambda = \sum _{i,j=1}^{n} \lambda _{ij}X_i Y_j={\,}^t X\Lambda Y, \] where Λ = ( λ i j ) \Lambda =(\lambda _{ij}) is a complex n × n n\times n matrix. Such operators never satisfy a cone condition in the sense of Sjöstrand and Hörmander. We may assume that P Λ \mathcal {P}_\Lambda cannot be viewed as a differential operator on a lower-dimensional Heisenberg group. Under the mild condition that Re ⁡ Λ , \operatorname {Re}\Lambda , Im ⁡ Λ \operatorname {Im}\Lambda and their commutator are linearly independent, we show that P Λ \mathcal {P}_\Lambda is not locally solvable, even in the presence of lower-order terms, provided that n ≥ 7 n\ge 7 . In the case n = 3 n=3 we show that there are some operators of the form described above that are locally solvable. This result extends to the Heisenberg group H 3 \mathbb {H}_3 a phenomenon first observed by Karadzhov and Müller in the case of H 2 . \mathbb {H}_2. It is interesting to notice that the analysis of the exceptional operators for the case n = 3 n=3 turns out to be more elementary than in the case n = 2. n=2. When 3 ≤ n ≤ 6 3\le n\le 6 the analysis of these operators seems to become quite complex, from a technical point of view, and it remains open at this time.