Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term
This article deals with regularity results for solutions of ultraparabolic equations. More precisely the following equation in \(\mathbb{R}^{N+1}\) is considered: \[ {\mathcal L}_vu:-{\mathcal L}_0u+{\mathcal V}u= 0\tag{1} \] where \({\mathcal L}_0\) is a linear second order operator of the form \[ {\mathcal L}_0=\sum^m_{k=1}X^2_k+X_0-\partial_t \tag{2} \] the terms \(X_k\) denote smooth vector fields on \(\mathbb{R}^N\) i.e. \[ X_k(x)= \sum^N_{j-1}a^k_j(x)\partial_{c_j}\;k=0,1,\dots,m.\tag{3} \] Finally \({\mathcal V}\) is a singular potential belonging to a Stummel-Kato class defined by the fundamental solution \(\Gamma_0\) of \({\mathcal L}_0\). The main result is an invariant Harnack inequality for positive solutions of (1) that extends previous results of A. Kogoj and E. Lanconelli for the solutions of the equation \({\mathcal L}_0u=0\). The proof, inspired by some arguments by Safonov, is based on pointwise lower bounds for the Green function \(G_0\) of \({\mathcal L}_0\) related to suitable ``cylindrical'' open sets. The Green function for the operator \({\mathcal L}_\nu\) is also constructed by the Levi parametrix method and \(L^p\) estimates and pointwise lower bounds are proved. An uniqueness result for the Cauchy-Dirichlet problem is also obtained.
scientific article; zbMATH DE number 5509272
scientific article