Liouville theorems for Fuchsian-type operators on the Heisenberg group
The paper deals with the equation (1) \(Lu=0\), where \[ Lu=- \sum^{2n}_{i,j=1} X^*_j \bigl(a_{ij} (x)X_iu+ d_j(x)u \bigr)+ \sum^{2n}_{i=1} b_i(x) X_iu+ c(x)u, \] \(X_j\) are the Heisenberg vector fields in \(\mathbb{R}^{2n+1}\), \(X^*_j\) is the \(L^2\)-adjoint of \(X_j\). The operator \(L\) is assumed to be uniformly subelliptic and weakly Fuchsian with respect to the intrinsic dilations. The main result of the authors is that there exists a unique (up to a constant) positive local solution of equation (1) in \(\mathbb{R}^{2n+1}\). They also prove that every bounded local solution of equation (1) in \(\mathbb{R}^{2n+1}\) is of constant sign.
scientific article; zbMATH DE number 1068004