Harnack inequalities for integro -differential operators

Abstract

In the first part of this dissertation, we consider the operator [special characters omitted] defined on C² ([special characters omitted]) functions by[special characters omitted]Under the assumption that the local part of the operator is uniformly elliptic and with suitable conditions on n( x, h), we establish a Harnack inequality for functions that are nonnegative and harmonic in a domain. We also show that the Harnack inequality can fail without suitable conditions on n(x, h). A regularity theorem for those nonnegative harmonic functions is also proved.

In the second part, we consider the Dirichlet form given by[special characters omitted]Under the assumption that the {a_ij } are symmetric and uniformly elliptic and with suitable conditions on the nonlocal part, we obtain upper and lower bounds on the heat kernel of the Dirichiet form. We also prove a Harnack inequality and a regularity theorem for functions that are harmonic with respect to [special characters omitted].