Uniform estimates for fundamental solutions associated with nonlocal Dirichlet forms (Q1916998)

The paper studies fundamental solutions of parabolic equations associated with pseudodifferential operators with non-continuous coefficients. The solutions can be obtained as transition functions of a Feller process associated with a Dirichlet form of non-local type. More precisely, the processes under consideration are stable type processes associated with measures of the form \[ K(t,dx,dy)=k(t,x,y)|x-y|^{-d-\alpha}dx dy, \] where \(k\) is a measurable function that is continuous with respect to the time variable and that satisfies \[ c_1\leq k(t,x,y)\leq c_2+c_3t^{-\gamma/\alpha}|x-y|^\gamma. \] The author proves a lower estimate and the Hölder continuity for the fundamental solution of the corresponding equation. The proof is based on moment bounds.