Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (4): 1223-1249.doi: 10.1007/s10473-021-0412-z

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AN UPBOUND OF HAUSDORFF'S DIMENSION OF THE DIVERGENCE SET OF THE FRACTIONAL SCHRÖDINGER OPERATOR ON Hs($\mathbb{R}^n$)

Dan LI1, Junfeng LI2, Jie XIAO3   

  1. 1. School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China;
    2. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China;
    3. Department of Mathematics and Statistics, Memorial University, St. John's NL A1C 5S7, Canada
  • Received:2020-04-14 Revised:2020-09-18 Online:2021-08-25 Published:2021-09-01
  • Contact: Junfeng Lis E-mail:junfengli@dlut.edu.cn
  • Supported by:
    Li Dan and Li Junfeng were supported by NSFC-DFG (11761131002) and NSFC (12071052). Xiao Jie was supported by NSERC of Canada (202979463102000).

Abstract: Given $n\geq2$ and $\alpha > \frac 12$, we obtained an improved upbound of Hausdorff's dimension of the fractional Schrödinger operator; that is, $$ \sup\limits_{f\in H^s(\mathbb{R}^n)}\dim _H\left\{x\in\mathbb{R}^n:\ \lim_{t\rightarrow0}e^{{\rm i}t(-\Delta)^\alpha}f(x)\neq f(x)\right\}\leq n+1-\frac{2(n+1)s}{n}%\ \ \text{under}\ \ \frac{n}{2(n+1)} < s\leq\frac{n}{2} $$ for $\frac{n}{2(n+1)} < s\leq\frac{n}{2}$.

Key words: The Carleson problem, divergence set, the fractional Schrödinger operator, Hausdorff dimension, Sobolev space

CLC Number: 

  • 42B37
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