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

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

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

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

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

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

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

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

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

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doi:10.1007/s10473-021-0412-z

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (4): 1223-1249.doi: 10.1007/s10473-021-0412-z

李丹¹, 李俊峰², 肖杰³

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

收稿日期:2020-04-14 修回日期:2020-09-18 出版日期:2021-08-25 发布日期:2021-09-01
通讯作者: Junfeng Lis E-mail:junfengli@dlut.edu.cn
作者简介:Dan LI,E-mail:danli@btbu.edu.cn;Junfeng LI,E-mail:junfengli@dlut.edu.cn;Jie XIAO,E-mail:jxiao@math.mun.ca
基金资助:
Li Dan and Li Junfeng were supported by NSFC-DFG (11761131002) and NSFC (12071052). Xiao Jie was supported by NSERC of Canada (202979463102000).

Dan LI¹, Junfeng LI², Jie XIAO³

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).

摘要： 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

$\frac{n}{2(n+1)} < s\leq\frac{n}{2}$ .

关键词: The Carleson problem, divergence set, the fractional Schrödinger operator, Hausdorff dimension, Sobolev space

Abstract: Given $n\geq2$ and $\alpha > \frac 12$ , we obtained an improved upbound of Hausdorff's dimension of the fractional Schrödinger operator; that is,

$\frac{n}{2(n+1)} < s\leq\frac{n}{2}$ .

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

中图分类号:

42B37

李丹, 李俊峰, 肖杰. AN UPBOUND OF HAUSDORFF'S DIMENSION OF THE DIVERGENCE SET OF THE FRACTIONAL SCHRÖDINGER OPERATOR ON H^s( $\mathbb{R}^n$ )[J]. 数学物理学报(英文版), 2021, 41(4): 1223-1249.

Dan LI, Junfeng LI, Jie XIAO. AN UPBOUND OF HAUSDORFF'S DIMENSION OF THE DIVERGENCE SET OF THE FRACTIONAL SCHRÖDINGER OPERATOR ON H^s( $\mathbb{R}^n$ )[J]. Acta mathematica scientia,Series B, 2021, 41(4): 1223-1249.

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