Articles

PARTIAL SCHAUDER ESTIMATES FOR A SUB-ELLIPTIC EQUATION

  • Na WEI ,
  • Yongsheng JIANG ,
  • Yonghong WU
Expand
  • 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 date: 2015-04-28

  Revised date: 2015-09-29

  Online published: 2016-06-25

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)".

Abstract

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.

Cite this article

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 . DOI: 10.1016/S0252-9602(16)30051-0

References

[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 W2,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

Outlines

/