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 \[T^{\alpha}f(t,x) = \int_0^t \int_{{\mathbb R}^d} P^{\alpha}(t-s,x-y)f(s,y) dyds,\] where the kernel is \[P^{\alpha}(t,x) = \int_{{\mathbb R}^d} e^{2\pi ix\cdot\xi} e^{-t|\xi|^\alpha} d\xi.\] The operator $T^{\alpha}$ maps from $L^{p}F_{s}^{p,q}$ to $L^{p}F_{s+\alpha/p}^{p,q}$ continuously. It has an application to a class of stochastic integro-differential equations of the type $du = -(-\Delta)^{\alpha/2} u dt + f dX_t$.