Title

Harnack inequalities for integro-differential operators

Date of Completion

January 2006

Keywords

Mathematics

Degree

Ph.D.

Abstract

In the first part of this dissertation, we consider the operator L defined on C2 ( Rd ) functions by Lfx =12 i,j=1

    d
aijx 62fx 6xi6xj +i=1
    d
bi x6fx 6xi +R d\0 fx+h -fx-1 h≤1h˙1 fxn x,hdh. 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 Ef,f =12 Rd i,j=1
    d
aijx 6fx 6xi6f x6xj dx +Rd ×Rd fy-fx 2Jx,y dxdy. 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 E .^