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

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

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

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).

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

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/Abstract

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, $$ \sup\limits_{f\in H^s(\mathbb{R}^n)}\dim _H\left\{x\in\mathbb{R}^n:\ \lim_{t\rightarrow0}e^{{\rm i}t(-\Delta)^\alpha}f(x)\neq f(x)\right\}\leq n+1-\frac{2(n+1)s}{n}%\ \ \text{under}\ \ \frac{n}{2(n+1)} < s\leq\frac{n}{2} $$ 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

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.

0
/ / 推荐 H^s($\mathbb{R}^n$)”的文章，特向您推荐。请打开下面的网址：http://121.43.60.238/sxwlxbB/CN/abstract/abstract16475.shtml' name="neirong"> H^s($\mathbb{R}^n$)'>

导出引用管理器 EndNote|Reference Manager|ProCite|BibTeX|RefWorks

链接本文: http://121.43.60.238/sxwlxbB/CN/10.1007/s10473-021-0412-z

http://121.43.60.238/sxwlxbB/CN/Y2021/V41/I4/1223

参考文献

[1] Cho C H, Ko H. A note on maximal estimates of generalized Schrödinger equation. arXiv:1809. 03246v1
[2] Sjögren P, Sjölin P. Convergence properties for the time-dependent Schrödinger equation. Ann Acad Sci Fenn Ser A I Math, 1989, 14(1):13-25
[3] Barceló J A, Bennett J, Carbery A, Rogers K M. On the dimension of divergence sets of dispersive equations. Math Ann, 2011, 349(3):599-622
[4] Žubrinić D. Singular sets of Sobolev functions. C R Math Acad Sci Paris, 2002, 334(7):539-544
[5] Du X M, Zhang R X. Sharp L2 estimate of Schrödinger maximal function in higher dimensions. arXiv:1805. 02775v1
[6] Carleson L. Some analytic problems related to statistical mechanics//Euclidean harmonic analysis (Proc Sem, Univ Maryland, College Park, Md, 1979). Lecture Notes in Math, 779. Berlin:Springer, 1980:5-45
[7] Bourgain J. Some new estimates on oscillatory integrals. Essays on Fourier Analysis in Honor of Elias M. Stein (Princeton, NJ, 1991), Princeton Math Ser, Vol 42. New Jersey:Princeton University Press, 1995:83-112
[8] Bourgain J. On the Schrödinger maximal function in higher dimension. Proc Steklov Inst Math, 2013, 280(1):46-60
[9] Bourgain J. A note on the Schrödinger maximal function. J Anal Math, 2016, 130:393-396
[10] Lee S. On pointwise convergence of the solutions to Schrödinger equations in $\mathbb{R}^2$. Int Math Res Not, 2006. Art ID 32597, 21 pp
[11] Miao C X, Yang J W, Zheng J Q. An improved maximal inequality for 2D fractional order Schrödinger operators. Studia Math, 2015, 230(2):121-165
[12] Moyua A, Vargas A, Vega L. Schrödinger maximal function and restriction properties of the Fourier transform. Internat Math Res Notices, 1996(16):793-815
[13] Sjölin P. Regularity of solutions to the Schrödinger equation. Duke Math J, 1987, 55(3):699-715
[14] Sjölin P. Nonlocalization of operators of Schrödinger type. Ann Acad Sci Fenn Math, 2013, 38(1):141-147
[15] Tao T, Vargas A. A bilinear approach to cone multipliers. II. Applications Geom Funct Anal, 2000, 10(1):216-258
[16] Vega L. E1 Multiplicador de Schrödinger, la Function Maximal y los Operadores de Restriccion(thesis). Madrid:Departamento de Matematicas, Univ Autónoma de Madrid, 1988
[17] Vega L. Schrödinger equations:pointwise convergence to the initial data. Proc Amer Math Soc, 1988, 102(4):874-878
[18] Dahlberg B E G, Kenig C E. A note on the almost everywhere behavior of solutions to the Schrödinger equation//Harmonic analysis (Minneapolis, Minn, 1981). Lecture Notes in Math, 908. Berlin-New York:Springer, 1982:205-209
[19] Du X M, Guth L, Li X C. A sharp Schrödinger maximal estimate in $\mathbb{R}^2$ . Ann of Math, 2017, 186(2):607-640
[20] Lucà R, Rogers K. A note on pointwise convergence for the Schrödinger equation. Math Proc Cambridge Philos Soc, 2019, 166(2):209-218
[21] Lucà R, Rogers K. Coherence on fractals versus pointwise convergence for the Schrödinger equation. Comm Math Phys, 2017, 351(1):341-359
[22] Du X M, Guth L, Li X C, Zhang R X. Pointwise convergence of Schrödinger solutions and multilinear refined Strichartz estimates. Forum Math Sigma, 2018, 6, e14, 18 pp
[23] Lucà R, Rogers K. Average decay for the Fourier transform of measures with applications. J Eur Math Soc, 2019, 21(2):465-506
[24] Cho C H, Lee S, Vargas A. Problems on pointwise convergence of solutions to the Schrödinger equation. J Fourier Anal Appl, 2012, 18(5):972-994
[25] Lee S, Rogers K. The Schrödinger equation along curves and the quantum harmonic oscillator. Adv Math, 2012, 229(3):1359-1379
[26] Cho Y, Ozawa T, Xia S. Remarks on some dispersive estimates. Commun Pure Appl Anal, 2011, 10(4):1121-1128
[27] Dinh V D. Strichartz estimates for the fractional Schrödinger and wave equations on compact manifolds without boundary. J Differential Equations, 2017, 263(12):8804-8837
[28] Bourgain J, Demeter C. The proof of the l² decoupling conjecture. Ann of Math, 2015, 182(1):351-389
[29] Tao T. A sharp bilinear restrictions estimate for paraboloids. Geom Funct Anal, 2003, 13(6):1359-1384

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($\mathbb{R}^n$)

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 4

[1]	周建丰, 谭忠. REGULARITY OF WEAK SOLUTIONS TO A CLASS OF NONLINEAR PROBLEM[J]. 数学物理学报(英文版), 2021, 41(4): 1333-1365.
[2]	郭常予, 向长林. REGULARITY OF P-HARMONIC MAPPINGS INTO NPC SPACES[J]. 数学物理学报(英文版), 2021, 41(2): 633-645.
[3]	Abdelkrim MOUSSAOUI, Jean VELIN. EXISTENCE AND BOUNDEDNESS OF SOLUTIONS FOR SYSTEMS OF QUASILINEAR ELLIPTIC EQUATIONS[J]. 数学物理学报(英文版), 2021, 41(2): 397-412.
[4]	戴济能, 周静云. GLEASON'S PROBLEM ON FOCK-SOBOLEV SPACES[J]. 数学物理学报(英文版), 2021, 41(1): 337-348.
[5]	Jae-Myoung KIM, Yun-Ho KIM, Jongrak LEE. RADIALLY SYMMETRIC SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING NONHOMOGENEOUS OPERATORS IN AN ORLICZ-SOBOLEV SPACE SETTING[J]. 数学物理学报(英文版), 2020, 40(6): 1679-1699.
[6]	陶文宇, 陈艳萍, 李纪黎. GRADIENT ESTIMATES FOR THE COMMUTATOR WITH FRACTIONAL DIFFERENTIATION FOR SECOND ORDER ELLIPTIC OPERATORS[J]. 数学物理学报(英文版), 2019, 39(5): 1255-1264.
[7]	程峰, 李维喜, 徐超江. GEVREY REGULARITY WITH WEIGHT FOR INCOMPRESSIBLE EULER EQUATION IN THE HALF PLANE[J]. 数学物理学报(英文版), 2017, 37(4): 1115-1132.
[8]	肖杰胜, 曹广福. WANDERING SUBSPACES OF THE HARDY-SOBOLEV SPACES OVER Dⁿ[J]. 数学物理学报(英文版), 2016, 36(5): 1467-1473.
[9]	张振亮, 曹春云. ON POINTS CONTAIN ARITHMETIC PROGRESSIONS IN THEIR LÜROTH EXPANSION[J]. 数学物理学报(英文版), 2016, 36(1): 257-264.
[10]	吴云顺, 谭忠. ASYMPTOTIC BEHAVIOR OF THE STOKES APPROXIMATION EQUATIONS FOR COMPRESSIBLE FLOWS IN R³[J]. 数学物理学报(英文版), 2015, 35(3): 746-760.
[11]	徐艳艳, 陈广贵, 甘莹, 许燕. PROBABILISTIC AND AVERAGE LINEAR WIDTHS OF SOBOLEV SPACE WITH GAUSSIAN MEASURE IN SPACE S_Q(T) (1≤Q≤∞)[J]. 数学物理学报(英文版), 2015, 35(2): 495-507.
[12]	傅仰耿, 刘正荣, 唐昊. NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE[J]. 数学物理学报(英文版), 2014, 34(6): 1781-1794.
[13]	陈振龙. ON INTERSECTIONS OF INDEPENDENT NONDEGENERATE DIFFUSION PROCESSES[J]. 数学物理学报(英文版), 2014, 34(1): 141-161.
[14]	王宏伟. LOCAL WELL-POSEDNESS IN SOBOLEV SPACES WITH NEGATIVE INDICES FOR A SEVENTH ORDER DISPERSIVE EQUATION[J]. 数学物理学报(英文版), 2014, 34(1): 199-208.
[15]	彭茹, 欧阳才衡. CARLESON MEASURES FOR BESOV-SOBOLEV SPACES WITH APPLICATIONS IN THE UNIT BALL OF Cⁿ[J]. 数学物理学报(英文版), 2013, 33(5): 1219-1230.