Local solvability for positive combinations of generalized sub-Laplacians on the Heisenberg group (Q2723510)

Let \(w\) denote the symplectic form on \(\mathbb{R}^{2n}\) given by NEWLINE\[NEWLINEw(z,z')= ^tz'Jz,\quad J=\left(\begin{matrix} 0 & I_n\\I_n & 0\end{matrix}\right).NEWLINE\]NEWLINE The Heisenberg group \(\mathbb{H}_n\) is \(\mathbb{R}^{2n}\times\mathbb{R}\), endowed with the group law \((z,u)(z',u') =(z + z', u + u' -\tfrac 12 w(z,z')).\) For \(z = (x,y) = (x_1,\cdots,x_n,y_1\cdots,y_n)\), a basis for the Lie algebra \({\mathfrak H}_n\) of \(\mathbb{H}_n\) is given by the left-invariant vector fields NEWLINE\[NEWLINEX_j=\frac{\partial}{\partial x_j}-\tfrac 12 y_j\;\frac{\partial}{\partial u},\quad j=1,\dots,n,\quad Y_j=\frac{\partial}{\partial y_j}+\tfrac 12 x_j\;\frac{\partial}{\partial u} ,\quad j=1,\dots,n,\quad V=\frac{\partial}{\partial u}.NEWLINE\]NEWLINE Let \(S\in\mathfrak {sp}(n,\mathbb{C})\) (the symplectic Lie algebra). We denote by \(A = (a_{jk})\) the symplectic matrix \(A = SJ\), and \(\Delta_S=\sum^{2n}_{j,k=1} a_{jk} V_jV_k\), where \(V_j = X_j\), \(V_{n+j}= Y_j\), \(j = 1,\dots, n\). The matrix \(S\) assumes a block diagonal form NEWLINE\[NEWLINES=\left(\begin{matrix} \gamma_1(S)_{(1)}\\ & \gamma_2(S)_{(2)}\\ &&\ddots\\ &&& \gamma_mS_{(m)}\end{matrix}\right)NEWLINE\]NEWLINE with respect to a suitable decomposition of \(\mathbb{R}^{2n}\) into symplectic subspaces under \(\gamma_j\in \mathbb{C}^\times =\mathbb{C}\setminus \{0\}\) and \(S^2_{(j)}=-I\), \(j =1,\cdots,n\). We may assume that \(S_{(j)}\) is either of the form of type NEWLINE\[NEWLINES_{(j)}=\left(\begin{matrix} i\varepsilon_j\lambda_j & \lambda^2_j-1\\1 & -i\varepsilon_j\lambda_j\end{matrix}\right)NEWLINE\]NEWLINE with \(\lambda_j\in \{-1\}\cup [0,\infty)\), and \(\varepsilon_j = 1\) if \(|j|\leq 1\) and \(\varepsilon_j=\pm 1\) if \(\lambda_j > 1\), or of the form of type 3 NEWLINE\[NEWLINES_{(j)}=\left(\begin{matrix} 0\;& i\\ i\;& 0\end{matrix}\right).NEWLINE\]NEWLINE Denote by \(\sigma_S\) the principal symbol of \(-\Delta_S\). We assume that \({\mathfrak R}\sigma_S\geq 0\). Then it follows that \(\Delta_S\), is of the form NEWLINE\[NEWLINE\Delta_S=\sum^r_{j=1}\gamma_j((1-\lambda^2_j)X^2_j+Y^2_j+i\lambda_j(X_jY_j+Y_jX_j))+i\sum^n_{j=r+1} \gamma_j(X_jY_j+Y_jX_j),NEWLINE\]NEWLINE where \(0\leq r\leq n\), \(|\lambda_j|\leq 1\), \(\gamma_j\in\mathbb{C}^\times\) for \(j=1,\dots,r\), \(\gamma_j>0\) for \(j=r+1,\dots,n\) and where for each \(j =1,\dots,r\) and every \(\xi_j,\eta_j\in\mathbb{R}\) NEWLINE\[NEWLINE{\mathfrak R}[\gamma_j((1-\lambda^2_j)\xi^2_j+\eta^2_j+2i\lambda_j\xi_j\eta_j)]\geq 0NEWLINE\]NEWLINE provided we choose appropriate coordinates. We put \(\Delta_{S,\alpha}=\Delta_S+i\alpha U\), \(\alpha\in \mathbb{C}\). Set NEWLINE\[NEWLINEE^\pm=\{\pm\sum^r_{j=1} \gamma_j(2l_j+1):l_1,\dots,l_r\in\mathbb{N}\}NEWLINE\]NEWLINE and put \(E = E^+\cup E^-\). We denote by \(n_1,n^+_2\) and \(n^-_2\) the number of type 1 blocks \(S_{(j)}\) with \(|\lambda_j|<1\), \(\lambda_j=1\) and \(\lambda_j=-1\), respectively, and by \(n_3\) the number of type 3 blocks. The solvability of operators of the form \(\Delta_{S,\alpha}\) has been studied by \textit{M. Peloso, F. Ricci} and \textit{D. Müller} [J. Reine Angew. Math. 513, 181-234 (1999; Zbl 0937.43003)]. It was shown that solvability holds for \(\alpha\notin E\). The case \(\alpha\in E\) remained open. The purpose of this paper is to close this gap. The results are the following: NEWLINENEWLINENEWLINETheorem. Assume \(\gamma_j > 0\) for \(j = 1,\dots,r\). Then \(E\) is discrete in \(\mathbb{C}\), and the following holds:NEWLINENEWLINENEWLINE(i) If \(\alpha\notin E\), then \(\Delta_{S,\alpha}\) is locally solvable.NEWLINENEWLINENEWLINE(ii) If \(\alpha\in E^-\), then \(\Delta_{S,\alpha}\) is locally solvable if and only if \(n^+_2+n_3>0\).NEWLINENEWLINENEWLINE(iii) If \(\alpha\in E^+\), then \(\Delta_{S,\alpha}\) is locally solvable if and only if \(n^-_2+n_3>0\) .NEWLINENEWLINENEWLINEThe results extend corresponding results of \textit{M. Peloso, F. Ricci} and \textit{D. Müller} [J. Funct. Anal. 148, 368-383 (1997; Zbl 0887.43004)].