PARTIAL SCHAUDER ESTIMATES FOR A SUB-ELLIPTIC EQUATION

doi:10.1016/S0252-9602(16)30051-0

数学物理学报(英文版) ›› 2016, Vol. 36 ›› Issue (3): 945-956.doi: 10.1016/S0252-9602(16)30051-0

PARTIAL SCHAUDER ESTIMATES FOR A SUB-ELLIPTIC EQUATION

魏娜¹, 蒋永生^1,2, 吴永洪^1,2

1. School of Statistic and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China;
2. Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

收稿日期:2015-04-28 修回日期:2015-09-29 出版日期:2016-06-25 发布日期:2016-06-25
通讯作者: Yongsheng JIANG,E-mail:jiangys@znufe.edu.cn E-mail:jiangys@znufe.edu.cn
作者简介:Na WEI,E-mail:weina@znufe.edu.cn;Yonghong WU,E-mail:Y.Wu@curtin.edu.au
基金资助:
This work was supported by the NSFC (11201486, 11326153). The first author was supported by "the Fundamental Research Funds for the Central Universities (31541411213)".

PARTIAL SCHAUDER ESTIMATES FOR A SUB-ELLIPTIC EQUATION

Na WEI¹, Yongsheng JIANG^1,2, Yonghong WU^1,2

1. School of Statistic and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China;
2. Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Received:2015-04-28 Revised:2015-09-29 Online:2016-06-25 Published:2016-06-25
Contact: Yongsheng JIANG,E-mail:jiangys@znufe.edu.cn E-mail:jiangys@znufe.edu.cn
Supported by:
This work was supported by the NSFC (11201486, 11326153). The first author was supported by "the Fundamental Research Funds for the Central Universities (31541411213)".

摘要/Abstract

摘要：

In this paper, we establish the partial Schauder estimates for the Kohn Laplace equation in the Heisenberg group based on the mean value theorem, the Taylor formula and a priori estimates for the derivatives of the Newton potential.

关键词: partial Schauder estimates, Kohn Laplace equation, Heisenberg group

Abstract:

Key words: partial Schauder estimates, Kohn Laplace equation, Heisenberg group

中图分类号:

35R05

魏娜, 蒋永生, 吴永洪. PARTIAL SCHAUDER ESTIMATES FOR A SUB-ELLIPTIC EQUATION[J]. 数学物理学报(英文版), 2016, 36(3): 945-956.

Na WEI, Yongsheng JIANG, Yonghong WU. PARTIAL SCHAUDER ESTIMATES FOR A SUB-ELLIPTIC EQUATION[J]. Acta mathematica scientia,Series B, 2016, 36(3): 945-956.

参考文献

[1] Arena G, Caruso A O, Causa G. Taylor formula on step two Carnot group. Rev Mat Iberoam, 2010, 26(1): 239-259
[2] Bonfiglioli A, Lanconellli E, Uguzzoni F. Stratified Lie Groups and Potential Theory for Their sub-Laplacians. Springer Monographs in Mathematics. Berlin: Springer, 2007
[3] Bramanti M, Brandolini L. Schauder estimates for parabolic nondivergence operators of Hörmander type. J Differential Equations, 2007, 234(1): 177-245
[4] Capogna L. Regularity of quasi-linear equations in the Heisenberg group. Comm Pure Appl Math, 1997, 50(09): 867-889
[5] Caffarelli L A. Interior a priori estimates for solutions of fully nonlinear equations. Ann Math, 1989, 130(2): 189-213
[6] Caffarelli L A. Interior W^2,p estimates for solutions of Monge-Ampère equations. Ann Math, 1990, 131(2): 135-150
[7] Capogna L, Han Q. Pointwise Schauder estimates for second order linear equations in Carnot groups. Proceedings for AMS-SIAM Harmonic Analysis Conference in Mt Holyhoke, 2001
[8] Douglis A, Nirenberg L. Interior estimates for elliptic systems of partial differential equations. Comm Pure Appl Math, 1955, 8: 503-538
[9] Dong H J, Kim S. Partial Schauder estimates for second order elliptic and parabolic equations. Calc Var Partial Differential Equations, 2011, 40(3/4): 481-500
[10] Jiang Y S, Tian F J. Schauder estimates for Kohn-Laplace equation in the Heisenberg group. Acta Math Sci, 2012, 32A(6): 1191-1198
[11] Fife P. Schauder estimates under incomplete Hölder continuity assumptions. Pacific J Math, 1963, 13: 511-550
[12] Folland G B. A fundamental solution for a subelliptic operator. Bull Amer Math Soc, 1973, 79: 373-376
[13] Folland G B, Stein E M. Estimates for the ∂_b complex and analysis on the Heisenberg group. Comm Pure Appl Math, 1974, 27: 429-522
[14] Folland G B. Subelliptic estimates and function spaces on nilpotent Lie groups. Ark Mat, 1975, 13: 161-207
[15] Garofalo N, Lanconelli E. Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Ann Inst Fourier (Grenoble), 1990, 40(2): 313-356
[16] Gaveau B. Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groups nilpotents. Acta Math, 1977, 139: 95-153
[17] Garofalo N, Tournier F. New properties of convex functions in the Heisenberg group. Trans Amer Math Soc, 2006, 358: 2011-2055
[18] Gutiérrez C, Lanconelli E. Schauder estimates for sub elliptic equations. J Evol Equ, 2009, 9(4): 707-726
[19] Safonov M V. The classical solution of the elliptic Bellman equation. Izv Akad Nauk SSSR Ser Mat, 1988, 52(6): 1272-1287; translation in Math USSR-Izv, 1989, 33(3): 597-612
[20] Tian G J, Wang X J. Partial regularity for elliptic equations. Discrete Contin Dyn Syst, 2010, 28(3): 899-913
[21] Uguzzoni F, Lanconelli E. On the Poisson kernel for the Kohn Laplacian. Rend Mat Appl (7), 1997, 17(4): 659-677
[22] Wang X J. Schauder estimates for elliptic and parabolic equations. Chin Ann Math Ser B, 2006, 27(6): 637-642

PARTIAL SCHAUDER ESTIMATES FOR A SUB-ELLIPTIC EQUATION

PARTIAL SCHAUDER ESTIMATES FOR A SUB-ELLIPTIC EQUATION

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 9

Metrics

本文评价

推荐阅读 10

[1]	杨乔华. ON CRITICAL CASES OF SOBOLEV'S INEQUALITIES FOR HEISENBERG GROUPS[J]. 数学物理学报(英文版), 2012, 32(4): 1584-1592.
[2]	李澎涛, 彭立中. L^p BOUNDEDNESS OF COMMUTATOR OPERATOR ASSOCIATED WITH SCHRÖDINGER OPERATORS ON HEISENBERG GROUP[J]. 数学物理学报(英文版), 2012, 32(2): 568-578.
[3]	江寅生. ENDPOINT ESTIMATES FOR FRACTIONAL INTEGRAL ASSOCIATED TO SCHRÖDINGER OPERATORS ON THE HEISENBERG GROUPS[J]. 数学物理学报(英文版), 2011, 31(3): 993-1000.
[4]	刘和平, 刘宇, 王海辉. MULTIRESOLUTION ANALYSIS, SELF-SIMILAR TILINGS AND HAAR WAVELETS ON THE HEISENBERG GROUP[J]. 数学物理学报(英文版), 2009, 29(5): 1251-1266.
[5]	原子霞, 钮鹏程. SEMILINEAR DEGENERATE HEAT INEQUALITIES WITH SINGULAR POTENTIAL ON THE HEISENBERG GROUP[J]. 数学物理学报(英文版), 2009, 29(2): 349-359.
[6]	栾静闻; 朱赋鎏. THE UNIFORMLY BOUNDEDNESS OF THE RIESZ TRANSFORMS ON THE CAYLEY HEISENBERG GROUPS[J]. 数学物理学报(英文版), 2008, 28(4): 895-914.
[7]	刘明菊; 陆善镇. A WEIGHTED ESTIMATE OF THE HORMANDER MULTIPLIER ON THE HEISENBERG GROUP[J]. 数学物理学报(英文版), 2006, 26(1): 134-144.
[8]	贾高. REGULARITIES AND SINGULARITIES OF THE ENERGY MINIMIZERS OF THE HEISENBERG GROUP TARGETS[J]. 数学物理学报(英文版), 2003, 23(4): 447-.
[9]	周广才, 郑维行. LOCAL HARDY SPACES ON HEISENBERG GROUP OVER LOCAL FIELDS[J]. 数学物理学报(英文版), 1996, 16(2): 129-141.