Please wait a minute...
北京大学期刊网 | 作者  审稿人  编委专家  工作人员

首页   |   关于   |   浏览   |   投稿指南   |   新闻公告
数学进展 - 2016, Vol. 45(6): 801-800
“纪念《数学进展》创刊六十周年”特约综述文章
最优光滑化估计与迹定理综述
A Survey on Optimal Smoothing Estimates and Trace Theorems

Neal Bez1, 杉本充2
Neal Bez1, Mitsuru Sugimoto2

1. 玉大学大学院理工学研究科数理电子情报部门玉, 338-8570, 日本;
2. 名古屋大学大学院多元 数理科学研究科, 名古屋市千种区不老町, 464-8602, 日本
1. Department of Mathematics, Graduate School of Science and Engineering, Saitama University, Saitama 338-8570, Japan;
2. Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan

收稿日期: 2016-04-01
出版日期: 2016-11-10
2016, Vol. 45(6): 801-800
DOI: 10.11845/sxjz.2016001a


PDF
[290 KB]
865
下载
407
浏览

引用导出
0
    /   /   推荐

摘要 本文对自由薛定谔方程解的一大类光滑化估计中最优常数和最优函数存在性问题、以及其对偶估计式的迹定理问题近年来的研究进展进行综述.在我们近期的研究成果(含部分与町原秀二和斋藤洋树合作的成果)之前, 除B. Simon和渡边一雄对部分特例得到的结果之外, 关于此问题的结果很有限.基于B. Walther在更一般情况下的结果、以及我们在同一研究脉络下 对经典调和分析中Funk-Hecke定理的创新应用,我们现在可以对最优常数相关的大量自然问题给出解答; 本综述中介绍了前述研究成果, 并阐述了证明背后的主要思路.本文还提及这一问题与带漂移项的薛定谔方程中初值问题适定性相关的沟—竹内猜想之间的联系, 并提出一些公开问题.
Abstract:We survey recent progress on the problem of obtaining the optimal constant and existence of maximizers for a broad class of smoothing estimates for the free Schr\"odinger propagator, together with trace theorems as their dual estimates. Before our recent work (partially in collaboration with S. Machihara and H. Saito), a limited collection of results were known, including certain special cases due to B. Simon and K. Watanabe. Based on a result of B. Walther in a rather general context and our own innovation in the same vein using the Funk-Hecke theorem from classical harmonic analysis, we are now able to answer a number of natural questions regarding such sharp estimates; we exhibit these in this survey article and expose the main idea behind the proofs. We will also mention a relation with the Mizohata-Takeuchi conjecture on the well-posedness of the Cauchy problem for Schr\"odinger equations with drift terms and some open problems are highlighted.
PACS:  O175.25  
[1] Barcelo, J.A., Ruiz, A. and Vega, L., Weighted estimates for the Helmholtz equation and some applications, J. Funct. Anal., 1997, 150(2), 356-382.
[2] Bennett, J., Bez, N., Carbery, A. and Hundertmark, D., Heat-flow monotonicity of Strichartz norms, Anal. PDE}, 2009, 2(2): 147-158.
[3] Bez, N., Machihara, S. and Sugimoto, M., Extremisers for the trace theorem on the sphere, Math. Res. Lett., 2016, 23(3): 633-647.
[4] Bez, N., Saito, H. and Sugimoto, M., Applications of the Funk-Hecke theorem to smoothing and trace estimates, Adv. Math., 2015, 285: 1767-1795.
[5] Bez, N. and Sugimoto, M., On the best constant and extremisers for some smoothing estimates, J. Anal. Math., in press.
[6] Bez, N. and Sugimoto, M., Optimal forward and reverse estimates of Morawetz and Kato-Yajima type with angular smoothing index, J. Fourier Anal. Appl., 2015, 21(2): 318-341.
[7] Bez, N. and Sugimoto, M., Some recent progress on sharp Kato-type smoothing estimates, In: Complex Analysis and Dynamical Systems VI: Part 1: PDE, Differential Geometry, Radon Transform (Agranovsky, M.L., Ben-Artzi, M., Galloway, G. et al. eds.) Contemp. Math., Vol. 653, 2015, 41-50.
[8] Carbery, A. and Soria, F., Pointwise Fourier inversion and localisation in $\R^n$, J. Fourier Anal. Appl., 1997, 3(special issue): 847-858.
[9] Constantin, P. and Saut, J.C., Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1988, 1(2): 413-439.
[10] Fang, D.Y. and Wang, C.B., Weighted Strichartz estimates with angular regularity and their applications, Forum Math., 2011, 23(1): 181-205.
[11] Foschi, D., Maximizers for the Strichartz inequality, J. Eur. Math. Soc.} ( JEMS}), 2007, 9(4): 739-774.
[12] Hundertmark, D. and Zharnitsky, V., On sharp Strichartz inequalities in low dimensions, Int. Math. Res. Not. IMRN}, 2006, 2006: Article ID 34080, 18 pages.
[13] Hoshiro, T., Mourre's method and smoothing properties of dispersive equations, Comm. Math. Phys., 1999, 202(2): 255-265.
[14] Kato, T., Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 1966, 162: 258-279.
[15] Kato, T. and Yajima, K., Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys., 1989, 1(4): 481-496.
[16] Keel, M. and Tao, T., Small data blow-up for semilinear Klein-Gordon equations, Amer. J. Math., 1999, 121(3): 629-669.
[17] Kenig, C.E., Ponce, G. and Vega, L., Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 1991, 40(1): 33-69.
[18] Mizohata, S., Schr\"odinger type equations, In: On the Cauchy Problem, Beijing: Science Press, 1985, 166-177.
[19] Ozawa, T. and Rogers, K.M., Sharp Morawetz estimates, J. Anal. Math., 2013, 121: 163-175.
[20] Ozawa, T. and Tsutsumi, Y., Space-time estimates for null gauge forms and nonlinear Schr\"odinger equations, Differential Integral Equations}, 1998, 11(2): 201-222.
[21] Ruzhansky, M. and Sugimoto, M., A smoothing property of Schr\"odinger equations in the critical case, Math. Ann., 2006, 335(3): 645-673.
[22] Ruzhansky, M. and Sugimoto, M., Trace theorems: critical cases and best constants, Proc. Amer. Math. Soc., 2015, 143(1): 227-237.
[23] Simon, B., Best constants in some operator smoothness estimates, J. Funct. Anal., 1992, 107(1): 66-71.
[24] Sj\"olin, P., Regularity of solutions to the Schr\"odinger equation, Duke Math. J., 1987, 55(3): 699-715.
[25] Strichartz, R.S., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 1977, 44(3): 705-714.
[26] Sugimoto, M., Global smoothing properties of generalized Schr\"odinger equations, J. Anal. Math., 1998, 76(1): 191-204.
[27] Takeuchi, J., On the Cauchy problem for some non-Kowalewskian equations with distinct characteristic roots, J. Math. Kyoto Univ., 1980, 20(1): 105-124.
[28] Walther, B.G., A sharp weighted $L^2$-estimate for the solution to the time-dependent Schr\"odinger equation, Ark. Mat., 1999, 37(2): 381-393.
[29] Walther, B.G., Regularity, decay, and best constants for dispersive equations, J. Funct. Anal., 2002, 189(2): 325-335.
[30] Watanabe, K., Smooth perturbations of the self-adjoint operator $\Delta^{\frac{\alpha}{2}, Tokyo J. Math., 1991, 14(1): 239-250.
[31] Vega, L., Schr\"odinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc., 1988, 102(4): 874-878.
[1] 樊自安, 吴庆华. 带有次临界或临界增长的分数阶Schrödinger-Poisson方程组非平凡解的存在性[J]. 数学进展, 2019, 48(3): 352-362.
[2] 欧乾忠. 关于共形海赛不等式不存在性的一个注记[J]. 数学进展, 2017, 46(1): 154-158.
[3] 薛洪涛. 带梯度项的半线性椭圆方程正整体解的存在性[J]. 数学进展, 2017, 46(1): 103-110.
[4] 谭沈阳,黄体仁, 张学华. Carnot群上双调和函数的均值定理[J]. 数学进展, 2016, 45(6): 932-938.
[5] 李鸿亮, 明平兵. 一个应变梯度有限元的新基函数[J]. 数学进展, 2016, 45(6): 955-960.
[6] 张占利. 一类椭圆型方程不适定问题的Meyer小波正则化方法[J]. 数学进展, 2016, 45(3): 390-402.
[7] 郭玉星, 江寅生. 薛定谔型椭圆方程解的正则性[J]. 数学进展, 2015, 44(6): 923-930.
Viewed
Full text


Abstract

Cited

  Discussed   
首页 · 关于 · 关于OA · 法律公告 · 收录须知 · 联系我们 · 注册 · 登录


© 2015-2017 北京大学图书馆 .