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


Author: Minsuk Yang
Journal: Proc. Amer. Math. Soc. 143 (2015), 2571-2578
MSC (2010): Primary 42B25, 26D10, 60H15
DOI: https://doi.org/10.1090/S0002-9939-2015-12523-3
Published electronically: January 21, 2015
MathSciNet review: 3326037
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Abstract: In this paper we prove a parabolic Triebel-Lizorkin space estimate for the operator given by

$\displaystyle T^{\alpha }f(t,x) = \int _0^t \int _{{\mathbb{R}}^d} P^{\alpha }(t-s,x-y)f(s,y) dyds,$

where the kernel is

$\displaystyle P^{\alpha }(t,x) = \int _{{\mathbb{R}}^d} e^{2\pi ix\cdot \xi } e^{-t\vert\xi \vert^\alpha } d\xi .$

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$.

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Additional Information

Minsuk Yang
Affiliation: School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro Dongdaemungu, Seoul 130-722, South Korea
Email: yangm@kias.re.kr

DOI: https://doi.org/10.1090/S0002-9939-2015-12523-3
Received by editor(s): January 30, 2014
Published electronically: January 21, 2015
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2015 American Mathematical Society