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An Upbound of Hausdorff’s Dimension of the Divergence Set of the Fractional SchröDinger Operator On Hs(ℝn)

Authors
  • Li, Dan1
  • Li, Junfeng2
  • Xiao, Jie3
  • 1 Beijing Technology and Business University, Beijing, 100048, China , Beijing (China)
  • 2 Dalian University of Technology, Dalian, 116024, China , Dalian (China)
  • 3 Memorial University, St. John’s, NL, A1C 5S7, Canada , St. John’s (Canada)
Type
Published Article
Journal
Acta Mathematica Scientia
Publisher
Springer-Verlag
Publication Date
Jun 01, 2021
Volume
41
Issue
4
Pages
1223–1249
Identifiers
DOI: 10.1007/s10473-021-0412-x
Source
Springer Nature
Keywords
License
Yellow

Abstract

Given n ≥ 2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha > \tfrac{1}{2}$$\end{document}, we obtained an improved upbound of Hausdorff’s dimension of the fractional Schrödinger operator; that is, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathop {\sup }\limits_{f \in {H^s}({\mathbb{R}^n})} {\dim _H}\left\{ {x \in {{\mathbb{R}^n}}:\;\mathop {\lim }\limits_{t \to 0} {e^{{\rm{i}}t{{( - \Delta )}^\alpha }}}f(x) \ne f(x)} \right\} \le n + 1 - {{2(n + 1)s} \over n}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tfrac{n}{{2(n + 1)}} < s \le \tfrac{n}{2}$$\end{document}

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