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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 {aij } 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].
