Heat kernels of second order complex elliptic operators and applications (Q1379609)

The authors study second-order elliptic operators \({\mathcal L}\) on \(\mathbb{R}^n\) in divergence form: \[ {\mathcal L}=- \partial_{x_i}(a_{ij}(x) \partial_{x_j}- a_{i,n+1}(x))+ a_{n+1, j}\partial_{x_j}+ a_{n+1,n+1}(x), \] where the coefficients \(a_{ij}\), \(1\leq i, j\leq n+ 1\), are assumed complex-valued, bounded and, at least, measurable. When the space-dimension is larger than 3, the coefficients are assumed Hölder continuous and an ellipticity assumption is made. The authors are mainly interested in estimates for the (Green-) kernel of the resolvent and in heat-kernel estimates of Gaussian type. Estimates for operators are in Sobolev \(L^p\)-type spaces, and the authors also study square roots of the operators under consideration. In particular they study (and solve, making suitable assumptions) the ``Kato-square-root problem'', which consists in establishing the domain of the maximal accretive square-root of \({\mathcal L}\).