On the space-like analyticity in the extension problem for nonlocal parabolic equations

Abstract

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 ( ∂ t − div x ⁡ ( B ( x ) ∇ x ) ) s (\partial _t - \operatorname {div}_x(B(x)\nabla _x))^s , 0 > s > 1 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 ) B(x) is a uniformly elliptic matrix-valued function with real-analytic entries.