ON THE SPACE-LIKE ANALYTICITY IN THE EXTENSION PROBLEM FOR NONLOCAL PARABOLIC EQUATIONS

In this note we give an elementary proof of the space-like real analyticity of solutions to a degenerate evolution problem that arises in the study of fractional parabolic operators of the type ( partial differential t - divx(B(x) backward difference x))s, 0 < s < 1. Our primary interest is in the so-called extension variable. We show that weak solutions that are even in such variable, are in fact real-analytic in the totality of the space variables. As an application of this result we prove the weak unique continuation property for nonlocal parabolic operators of the type above, where B(x) is a uniformly elliptic matrix-valued function with real-analytic entries.