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 \[T^αf(t,x) = \int_0^t \int_{{\mathbb R}^d} P^α(t-s,x-y)f(s,y) dyds,\] where the kernel is \[P^α(t,x) = \int_{{\mathbb R}^d} e^{2πix\cdotξ} e^{-t|ξ|^α} dξ.\] 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 $du=-(-\Delta {)}^{\alpha /2}udt+fd{X}_t$.

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