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

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}