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Gradient Estimates for the Commutator with Fractional Differentiation for Second Order Elliptic Perators

Authors
  • Tao, Wenyu1
  • Chen, Yanping1
  • Li, Jili1
  • 1 University of Science and Technology Beijing, Department of Applied Mathematics, School of Mathematics and Physics, Beijing, 100083, China , Beijing (China)
Type
Published Article
Journal
Acta Mathematica Scientia
Publisher
Springer Singapore
Publication Date
Jul 10, 2019
Volume
39
Issue
5
Pages
1255–1264
Identifiers
DOI: 10.1007/s10473-019-0505-y
Source
Springer Nature
Keywords
License
Yellow

Abstract

Let L = −div(A∇) be a second order divergence form elliptic operator, where A is an accretive, n×n matrix with bounded measurable complex coefficients on ℝn. Let Lα2(0<α<1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\frac{\alpha}{2}}(0<\alpha<1)$$\end{document} denotes the fractional differential operator associated with L and (−Δ)α2b∈Ln/α(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta)^{\frac{\alpha}{2}}b\;\in\;L^{n/\alpha}(\mathbb{R}^n)$$\end{document}. In this article, we prove that the commutator [b, Lα2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\frac{\alpha}{2}}$$\end{document}] is bounded from the homogenous Sobolev space L˙α2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{L}_\alpha^2$$\end{document} (ℝn) to L2(ℝn).

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