In the first part of this dissertation, we consider the operator [Special characters omitted.] defined on C 2 ([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.] .
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