A parabolic Triebel–Lizorkin space estimate for the fractional Laplacian operator

Abstract:

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 $du = -(-\Delta )^{\alpha /2} u dt + f dX_t$.