Journal of Guangxi Normal University(Natural Science Edition) ›› 2010, Vol. 28 ›› Issue (1): 18-22.

Previous Articles     Next Articles

Sharp Hardy Type Inequalities for a Class of Degenerate Subelliptic Operator and Applications

WANG Sheng-jun1, DOU Jing-bo2   

  1. 1. Department of Mathematics and Information Sciences,QinghaiNormal University,Xining Qinghai 810008,China;
    2. School of Statistics,Xi'an Institute of Finance and Economics,Xi'an Shaanxi 710100,China
  • Received:2009-12-17 Online:2010-03-20 Published:2023-02-07

Abstract: A class of sharp Hardy type inequalityfor generalized Baouendi-Grushin operators is presented by improving the methods of the Euclid space.Then,a class of Hardy-Sobolev type inequality is established by its results.As applications,the positive property and the unbounded property from below for p-degenerate subelliptic operator are discussed.

Key words: generalized Baouendi-Grushin operators, regularization, sharp Hardy type inequality, Hardy-Sobolev type inequality

CLC Number: 

  • O175.2
[1] BAOUENDI M S.Sur une classe d'opérateurs elliptiques dégénérés[J].Bull Soc Math France,1967,95:45-87.
[2] GRUSIN V V.On a class of hypoelliptic operators[J].Math USSR Sbornik,1970,12(3):458-476.
[3] GAROFALO N.Unique continuation for a class of elliptic operatorswhich degenerate on a manifold of arbitrary codimension[J].J Differential Equations,1993,104(1):117-146.
[4] GAROFALS N,NHIE D M.Isoperimetric and Sobolev inequalities for Carnot-Caratheodory spaces and the existence of minimum surfaces[J].Comm PureAppl Math,1996,49(10):1081-1144.
[5] 黄启亮.关于p=3/2的Hardy不等式的一个加强改进[J].广西师范大学学报:自然科学版,2000,18(1):38-41.
[6] GARCIA A J,PERAL I.Hardy inequality and some critical elliptic and parabolic problems[J].J Diff Eqs,1998,144(2):441-476.
[7] 窦井波,韩亚洲.广义Baouendi-Grushin向量场上抛物型不等方程弱解的不存在性[J].郑州大学学报:理学版,2005,37(4):1-8.
[8] 张慧清,钮鹏程.关于一类向量场的Picone恒等式和Hardy不等式[J].数学杂志,2003,23(1):121-125.
[9] DOU Jing-bo,NIU Peng-cheng.Hardy inequalities and applications related to generalized Baouendi-Grushin vector fields[J].Advances in Mathematics,2008,37(3):321-331.
[10] D'AMBROZIO L.Some Hardy inequalities on the Heisenberg group[J].Differential Equations,2004,40(4):552-564.
[11] MONTI R,MORBIDELLI D.Kelvin transform for Grushin operators andcritical semilinear equations[J].Duke Math J,2006,131(1):167-202.
[12] NIU Peng-cheng,DOU Jing-bo.Hardy-Sobolev type inequalities for generalized Baouendi-Grushin operators[J].Miskolc Mathematical Notes,2007,8(1):73-87.
[1] DU Jinfeng, WANG Hairong, LIANG Huan, WANG Dong. Progress of Cross-modal Retrieval Methods Based on Representation Learning [J]. Journal of Guangxi Normal University(Natural Science Edition), 2022, 40(3): 1-12.
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
[1] CHEN Yong-qi, BAI Ke-zhao, KUANG hua, KONG Ling-jiang, LIU Mu-ren. Effect of Internal Layout on the Pedestrian Evacuation in the Classroom[J]. Journal of Guangxi Normal University(Natural Science Edition), 2011, 29(1): 1 -4 .
[2] XU Lun-hui, YE Fan. Acceleration Noise Model Based on Horizontal,Vertical and LateralAcceleration[J]. Journal of Guangxi Normal University(Natural Science Edition), 2011, 29(1): 5 -9 .
[3] YANG Li, KONG Ling-jiang. Capillary Force between Microparticles[J]. Journal of Guangxi Normal University(Natural Science Edition), 2012, 30(1): 1 -4 .
[4] HE Qing, LIU Jian, WEI Lianfu. Single-Photon Detectors as the Physical Limit Detections of Weak Electromagnetic Signals[J]. Journal of Guangxi Normal University(Natural Science Edition), 2022, 40(5): 1 -23 .
[5] BAI Ke-zhao, LUO Xu-dong, KONG Ling-jiang, LIU Mu-ren. Cellular Automaton Model of Date Transmission with Open Boundary Condition[J]. Journal of Guangxi Normal University(Natural Science Edition), 2010, 28(3): 1 -4 .
[6] XU Lun-hui, LIAO Ran-kun. Signal Phasing-Sequence Optimization of Intersection Based on Traffic Track[J]. Journal of Guangxi Normal University(Natural Science Edition), 2010, 28(3): 5 -9 .
[7] WANG Xiu-xin, QIN Li-mei, NONG Jing-hui, LIANG Zong-jin, ZHU Qi-jiang. Land Surface Temperature Retrieval with Mono-window Algorithm in Karst City[J]. Journal of Guangxi Normal University(Natural Science Edition), 2010, 28(3): 10 -14 .
[8] LI Yu-fang, ZHANG Jun-jian. Strong Consistency of the Regression Weighted Function Estimator for Negatively Associated Samples[J]. Journal of Guangxi Normal University(Natural Science Edition), 2010, 28(3): 15 -19 .
[9] JIA Bao-hua. A Strictly Stationary Associated Random Sequence Which Unsatisfythe Central Limit Theorem[J]. Journal of Guangxi Normal University(Natural Science Edition), 2010, 28(3): 20 -23 .
[10] CHEN Cui-ling, LI Ming, LIANG Jia-mei, LI Lüe. A Class of New Conjugate Gradient Method and Its Convergence Property Under the Wolfe Line Search[J]. Journal of Guangxi Normal University(Natural Science Edition), 2010, 28(3): 24 -28 .