IMPROVED GAGLIARDO-NIRENBERG INEQUALITIES ON HEISENBERG TYPE GROUPS

doi:10.1016/S0252-9602(11)60344-5

数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (4): 1583-1590.doi: 10.1016/S0252-9602(11)60344-5

IMPROVED GAGLIARDO-NIRENBERG INEQUALITIES ON HEISENBERG TYPE GROUPS

罗光洲

School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China

收稿日期:2008-11-27 修回日期:2009-12-18 出版日期:2011-07-20 发布日期:2011-07-20
基金资助:
This work was supported by National Science Foundation of China (10771175).

IMPROVED GAGLIARDO-NIRENBERG INEQUALITIES ON HEISENBERG TYPE GROUPS

LUO Guang-Zhou

School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China

Received:2008-11-27 Revised:2009-12-18 Online:2011-07-20 Published:2011-07-20
Supported by:
This work was supported by National Science Foundation of China (10771175).

摘要/Abstract

摘要：

Motivated by the idea of M. Ledoux who brings out the connection between Sobolev embeddings and heat kernel bounds, we prove an analogous result for Kohn’s sub-Laplacian on the Heisenberg type groups. The main result includes features of an inequality of either Sobolev or Galiardo-Nirenberg type.

关键词: Heisenberg type group, heat kernel, Sobolev inequality, Galiardo-Nirenberg inequality

Abstract:

Key words: Heisenberg type group, heat kernel, Sobolev inequality, Galiardo-Nirenberg inequality

中图分类号:

22E25

罗光洲. IMPROVED GAGLIARDO-NIRENBERG INEQUALITIES ON HEISENBERG TYPE GROUPS[J]. 数学物理学报(英文版), 2011, 31(4): 1583-1590.

LUO Guang-Zhou. IMPROVED GAGLIARDO-NIRENBERG INEQUALITIES ON HEISENBERG TYPE GROUPS[J]. Acta mathematica scientia,Series B, 2011, 31(4): 1583-1590.

参考文献

[1] Bakry D, Baudoin F, Bonnefont M, Chafai D, On gradient bounds for the heat kernel on the Heisenberg group. J Funct Anal, 2008, 255(8): 1905–1938

[2] Beals R, Gaveau B, Greiner P, Hamilton-Jacobi theory and the heat kernel on Heisenberg groups. J Math Pures Appl, 2000, 79: 633–689

[3] Bonfiglioli A, Uguzzoni F. Nonlinear Liouville theorems for some critical problems on H-type groups. J Funct Anal, 2004, 207: 161–215

[4] Cohen A, Dahmen W, Daubechies I, DeVore R. Harmonic analysis of the space BV. Rev Mat Iberoamer-icana, 2003, 19: 235–263

[5] Hsiao L, Li H L. Compressible Navier-Stokes-Poisson equations. Acta Math Sci, 2010, 30B(6): 1937–1948

[6] Eldredge N. Precise estimates for the subelliptic heat kernel on H-type groups. J Math Pures Appl, 2009, 92(1): 52–85

[7] Eldredge N. Gradient estimates for the subelliptic heat kernel on H-type groups. J Funct Anal, 2010, 258(2): 504–533

[8] Folland G. Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Math., 1975, 13: 161–207

[9] Gaveau B. Principe de moindre action, propagation de la chaleur et estim´ees sous elliptiques sur certains groupes nilpotents. Acta Math, 1977, 139: 95–153

[10] Kaplan A. Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms. Trans Amer Math Soc, 1980, 258(1): 147–153

[11] Ledoux M. On improved Sobolev embedding theorems. Math Res Lett, 2003, 10: 659–669

[12] Li H Q. Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg. J Funct Anal, 2006, 236(2): 369–394

[13] Li H Q. Estimations asymptotiques du noyau de la chaleur sur les groupes de Heisenberg. C R Acad Sci Paris, Ser I, 2007, 344(8): 497–502

[14] Li Aobing, Li Yanyan. A Harnack type inequality for some conformally in variant equations on half Euclidean space. Acta Math Sci, 2009, 29B(4): 1105–1112

[15] Randall J. The heat kernel for generalized Heisenberg groups. J Geom Anal, 1996, 6(2): 287–316

[16] N. Varopoulos, Analysis on nilpotent Lie groups. J Funct Anal, 1986, 66: 406–431

[17] Yang Q, Zhu F. The heat kernel on H-type groups. Proc Amer Math Soc, 2008, 136(4): 1457–1464

IMPROVED GAGLIARDO-NIRENBERG INEQUALITIES ON HEISENBERG TYPE GROUPS

IMPROVED GAGLIARDO-NIRENBERG INEQUALITIES ON HEISENBERG TYPE GROUPS

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