Spectral theory of singular elliptic operators with measurable coefficients (Q1266254)

Consider a selfadjoint operator on \(L^2(\Omega,b dx),\) given formally by \[ Hf=(-1)^m b^{-1}\sum_{| \alpha| =m,| \beta| =m} D^\alpha (b a_{\alpha, \beta} D^\beta f) \] with Dirichlet boundary conditions. Here \(\Omega \subset {\mathbb R}^N\), \(2m>N, \) the weight \(b\) and the coefficients \(a_{\alpha,\beta}\) are positive, measurable functions, satisfying the superellipticity condition, \(c^{-1}a(x)| p| ^2\leq \sum a_{\alpha,\beta}(x)p_\alpha \bar{p}_ \beta \leq c a(x)| p| ^2\), \(p\in {\mathbb C}^\nu\), as well as some weighted embedding estimates. It is proven that the heat kernel admits Gaussian type estimates. In particular, the corresponding \(L^p\)-spectrum is \(p\)-independent.