We consider the Dirichlet form given by E(f,f)=12∫Rd∑i,j=1daij(x)∂f(x)∂xi∂f(x)∂xjdx E(f,f)=12∫Rd∑i,j=1daij(x)∂f(x)∂xi∂f(x)∂xjdx +∫Rd×Rd(f(y)−f(x))2J(x,y)dxdy. +∫Rd×Rd(f(y)−f(x))2J(x,y)dxdy. Under the assumption that the aij are symmetric and uniformly elliptic and with suitable conditions on J, the nonlocal part, we obtain upper and lower bounds on the heat kernel of the Dirichlet form. We also prove a Harnack inequality and a regularity theorem for functions that are harmonic with respect to E.
- Harnack inequality
- heat kernel
- Hölder continuity
- integro-differential operators