A remarkable class of second order differential operators on the Heisenberg group \(\mathbb{H}_2\) (Q5947243)

Let \(L\) be a second order complex coefficient operator on the Heisenberg group \(\mathbb{H}_2\) of the form \[ L = \sum^2_{j,k=1}\alpha_{jk}X_jY_k + \sum^2_{k=1}(a_kX_k+ b_kY_k) + cU + d), \] where \(X_j = \partial x_j\), \(Y_j = \partial_{y_j} +x_j\partial_u\), \(U = \partial_u\), and where the coefficients \(\alpha_{jk}\), \(a_k\), \(b_k\), \(c\), \(d\) are complex-valued smooth functions of \((x, y, u) = (x_1, x_2, y_1,y_2,u)\) in a bounded open neighborhood \(V\) of the origin in \(\mathbb{R}^5\). In this article the authors study the problem of local solvability of these operators \(L\). Let \(A\) and \(B\) be the real part and the imaginary part of the matrix \({\mathcal A} = (\alpha_{jk})\), respectively, i.e. \({\mathcal A} = A + iB\). Let \(E\) be a symmetric matrix of the form \[ E={}^tAJB-{}^tBJA, \] where \[ J=\left(\begin{smallmatrix} 0 & 1\\ 1 & 0\end{smallmatrix}\right). \] Theorem 1. Assume that, at the origin, \(\mathcal A\) satisfies the assumption \[ \text{det }E<0\tag{\(H_-\)} \] and that the commutator \([A(0),B(0)]\) does not vanish. Then the operator \(L\) is not locally solvable at the origin. Theorem 2. Assume that, at the origin, \(\mathcal A\) satisfies the assumption \[ \text{det }E > 0.\tag{\(H_+\)} \] Then there exist complex numbers \(\mu\) and \(\nu\) depending only on the coefficients of the matrix \({\mathcal A}(0)\), such that the following holds: (i) If \[ -(i\alpha + 1)\nu + (c(0) + \text{tr }{\mathcal A})\mu\not\in\mathbb{Z}\tag{1} \] for every \(\alpha\in\mathbb{R}\), then the transposed operator \(^tL\) is locally solvable at the origin. Moreover, there exists a neighborhood \(V_1\subset V\) of the origin such that the following a priori estimate is valid: \[ \|X_jY_k\phi\|+ \|Y_j X_k\phi\|\leq C\|L\phi\|\tag{2} \] for every \(\phi\in C^\infty_0(V_1)\) and \(j, k = 1,2\), and \(\|\cdot \|\) stands for the \(L^2\)-norm. (ii) If (1) does not hold, then (2) fails to be true. However, \(^tL\) is still locally solvable, provided all coefficients \(\alpha_{jk},a_k,b_k,c,d\) are constant, i.e. that \(L\) is left-invariant on \(\mathbb{H}_2\), and that \((a_1,a_2)\in{\mathcal A}(\mathbb{C}^2)\), \((b_1,b_2)\in {}^t{\mathcal A}(\mathbb{C}^2)\). Clearly, the latter condition holds if \(\text{det }{\mathcal A}\neq 0\). The following proposition complements Theorem 2. Proposition. (i) If \(\mathcal A\) satisfies condition \((H_+)\), then \(\nu\) is of the form \(\nu=\tfrac 12 \text{tr }{\mathcal A}\mu+\rho,\) where \(\rho\in \{-1,0,1\}\). Moreover, \(\text{Re} \nu\neq 0\). -- Thus (1) is equivalent to \[ \left(c(0)-(i\alpha-1)\;\frac{\text{tr} {\mathcal A}}{2}\right)\mu-i\alpha \rho\not\in\mathbb{Z} \] for every \(\alpha\in\mathbb{R}\). Moreover, either \(\rho = 0\) and \(\mu\neq 0\), or \(\mu = 0\) and \(\rho\neq 0\). (ii) If \(\mu= 0\), then (1) does not hold for any value of \(c(0)\). (iii) If \(\mu\neq 0\), then (1) holds if and only if \(c(0)\notin\frac\nu\mu-\text{tr }{\mathcal A}+\frac{i\nu}{\mu} \mathbb{R}+\frac{1}{\mu} \mathbb{Z},\) hence for generic \(c(0)\in\mathbb{C}\). Moreover, if \(\Delta=(\text{tr }{\mathcal A})^2-4\text{det }{\mathcal A}\) denotes the discriminant of the characteristic polynomial of \(\mathcal A\), then \(\Delta\neq 0\), so that one may define: \(N=\text{tr }{\mathcal A}/\sqrt{\Delta}\). Then \(\nu=\pm N\) and \(\mu=\pm 2/\sqrt{\Delta}\). In particular, \(\text{Re } N\neq 0\), so that \(\kappa = \frac{|N|^2+\text{Re }(\overline{Nc(0)\mu})}{\text{Re }N}\) is well defined. Then (1) holds if and only if \(\kappa\not\in\mathbb{Z}\). As a corollary to the results, the authors prove that the exceptional operator, which arose in [\textit{D. Müller, M. M. Peloso} and \textit{F. Ricci}, J. Reine Angew. Math. 513, 181-234 (1999; Zbl 0937.43003)], is locally solvable.