Finite propagation speed and kernels of strictly elliptic operators (Q1066459)

Resolvent and semigroup kernels are constructed and estimated in \(L^ p\)-spaces for a class of global elliptic operators on \({\mathbb{R}}^ n\) of the form \(A=\rho A_ 0+B\), where \(A_ 0=\sum_{| \alpha | =m}a_{\alpha}(x)D^{\alpha}\) is uniformly elliptic of order m, \(\rho\) satisfies a ''finite propagation speed'' condition, \(\rho (x)=O(| x|^ m)\), and perturbation \(B=\sum_{| \alpha | \leq m}b_{\alpha}D^{\alpha}\) may have \(L^ p\)-singular coefficients. This class is modeled after Schrödinger-type operators: \(A=-\nabla \cdot \rho \nabla +B\cdot \nabla +V\). Applications are given to closedness and essential selfadjointness of A, its \(L^ p\)-domains, ''resolvent summability'' (i.e. \(L^ p\)-convergence: \(\zeta (\zeta -A)^{-1}f\to f\) as \(\zeta\) \(\to \infty)\), and existence of a holomorphic semigroup \(\{e^{-tA}\}Re t>0\). Most results are valid in \(L^ p\)-spaces, \(1<p<\infty.\) The basic techniques of the paper fail in the limiting case \(p=1\). The ensuring problems and examples are briefly discussed at the end.