A PARABOLIC TRIEBEL-LIZORKIN SPACE ESTIMATE FOR THE FRACTIONAL LAPLACIAN OPERATOR

In this paper we prove a parabolic Triebel-Lizorkin space estimate for the operator given by T-alpha f(t, x) = integral(t)(0) integral(Rd) P-alpha (t - s, x - y) f (s, y)dyds, where the kernel is P-alpha (t, x) - integral(Rd) e(2 pi ix.xi e) (-t vertical bar xi vertical bar alpha) d xi. The operator T-alpha maps from (LFsp, q)-F-p to LpFs+alpha/pp, q continuously. It has an application to a class of stochastic integro- differential equations of the type du = -(-Delta)(alpha/2)udt + fdX(t).