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

TL;DR

This paper establishes a parabolic Triebel-Lizorkin space estimate for a fractional Laplacian operator, demonstrating its boundedness and applying it to certain stochastic integro-differential equations.

Contribution

It provides a novel Triebel-Lizorkin space estimate for the fractional Laplacian operator and explores its implications for stochastic integro-differential equations.

Findings

The operator maps between specific Triebel-Lizorkin spaces continuously.

The estimate facilitates analysis of stochastic integro-differential equations.

Application to equations involving fractional Laplacian and stochastic processes.

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