Advanced Search
Article Contents
Article Contents

A general approach to weighted $L^{p}$ Rellich type inequalities related to Greiner operator

  • * Corresponding author

    * Corresponding author
Abstract Full Text(HTML) Related Papers Cited by
  • In this paper we exhibit some sufficient conditions that imply general weighted $L^{p}$ Rellich type inequality related to Greiner operator without assuming a priori symmetric hypotheses on the weights. More precisely, we prove that given two nonnegative functions $a$ and $b$, if there exists a positive supersolution $\vartheta $ of the Greiner operator $Δ _{k}$ such that

    ${\Delta _k}\left( {a|{\Delta _k}\vartheta {|^{p - 2}}{\Delta _k}\vartheta {\rm{ }}} \right) \ge b{\vartheta ^{p - 1}}$

    almost everywhere in $\mathbb{R}^{2n+1}, $ then $a$ and $b$ satisfy a weighted $L^{p}$ Rellich type inequality. Here, $p>1$ and $Δ _{k} = \sum\nolimits_{j = 1}^n {} \left(X_{j}^{2}+Y_{j}^{2}\right) $ is the sub-elliptic operator generated by the Greiner vector fields

    ${X_j} = \frac{\partial }{{\partial {x_j}}} + 2k{y_j}|z{{\rm{|}}^{2k - 2}}\frac{\partial }{{\partial l}}, \;\;\;\;{Y_j} = \frac{\partial }{{\partial {y_j}}} - 2k{x_j}|z{|^{2k - 2}}\frac{\partial }{{\partial l}}, \;\;\;\;j = 1, ..., n, $

    where $\left( z, l\right) = \left( x, y, l\right) ∈\mathbb{R}^{2n+1} = \mathbb{R}^{n}×\mathbb{R}^{n}×\mathbb{R}, $ $|z{\rm{|}} = \sqrt {\sum\nolimits_{j = 1}^n {} \left( {x_j^2 + y_j^2} \right)} $ and $k≥ 1$. The method we use is quite practical and constructive to obtain both known and new weighted Rellich type inequalities. On the other hand, we also establish a sharp weighted $L^{p}$ Rellich type inequality that connects first to second order derivatives and several improved versions of two-weight $L^{p}$ Rellich type inequalities associated to the Greiner operator $Δ _{k}$ on smooth bounded domains $Ω $ in $\mathbb{R}^{2n+1}$.

    Mathematics Subject Classification: Primary: 26D10, 22E30; Secondary: 43A80.


    \begin{equation} \\ \end{equation}
  • 加载中
  •   Adimurthi , M. Grossi  and  S. Santra , Optimal Hardy-Rellich inequalities, maximum principle and related eigenvalue problem, J. Funct. Anal., 240 (2006) , 36-83.  doi: 10.1016/j.jfa.2006.07.011.
      Adimurthi  and  S. Santra , Generalized Hardy-Rellich inequalities in critical dimension and its applications, Commun. Contemp. Math., 11 (2009) , 367-394.  doi: 10.1142/S0219199709003405.
      S. Ahmetolan  and  I. Kombe , A sharp uncertainty principle and Hardy-Poincaré inequalities on sub-Riemannian manifolds, Math. Inequal. Appl., 15 (2012) , 457-467.  doi: 10.7153/mia-15-40.
      S. Ahmetolan  and  I. Kombe , Hardy and Rellich type inequalities with two weight functions, Math. Inequal. Appl., 19 (2016) , 937-948.  doi: 10.7153/mia-19-68.
      W. Allegretto  and  Y. X. Huang , A Picone's identity for the p−Laplacian and applications, Nonlinear Anal., 32 (1998) , 819-830.  doi: 10.1016/S0362-546X(97)00530-0.
      G. Barbatis , Improved Rellich inequalities for the polyharmonic operator, Indiana Univ. Math. J., 55 (2006) , 1401-1422.  doi: 10.1512/iumj.2006.55.2752.
      R. Beals , B. Gaveau  and  P. Greiner , On a geometric formula for the fundamental solution of sub-elliptic Laplacians, Math. Nachr., 181 (1996) , 81-163.  doi: 10.1002/mana.3211810105.
      R. Beals , B. Gaveau  and  P. Greiner , Uniform hypoelliptic Green's functions, J. Math. Pures Appl., 77 (1998) , 209-248.  doi: 10.1016/S0021-7824(98)80069-X.
      D. M. Bennett , An extension of Rellich's inequality, Proc. Amer. Math. Soc., 106 (1989) , 987-993.  doi: 10.2307/2047283.
      P. Caldiroli  and  R. Musina , Rellich inequalities with weights, Calc. Var. Partial Differential Equations, 45 (2012) , 147-164.  doi: 10.1007/s00526-011-0454-3.
      E. B. Davies  and  A. M. Hinz , Explicit constants for Rellich inequalities in Lp(Ω), Math. Z., 227 (1998) , 511-523.  doi: 10.1007/PL00004389.
      A. Detalla , T. Horiuchi  and  H. Ando , Sharp remainder terms of the Rellich inequality and its application, Bull. Malays. Math. Sci. Soc., 35 (2012) , 519-528. 
      G. B. Folland , A fundamental solution for a sub-elliptic operator, Bull. Amer. Math. Soc., 79 (1973) , 373-376.  doi: 10.1090/S0002-9904-1973-13171-4.
      V. A. Galaktionov  and  I. V. Kamotski , On nonexistence of Baras-Goldstein type for higherorder parabolic equations with singular potentials, Trans. Am. Math. Soc., 362 (2010) , 4117-4136.  doi: 10.1090/S0002-9947-10-04855-5.
      N. Garofalo  and  E. Lanconelli , Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann Inst Fourier (Grenoble), 40 (1990) , 313-356. 
      F. Gazzola , H. C. Grunau  and  E. Mitidieri , Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc., 356 (2004) , 2149-2168.  doi: 10.1090/S0002-9947-03-03395-6.
      P. C. Greiner , A fundamental solution for a nonelliptic partial differential operator, Canad. J. Math., 31 (1979) , 1107-1120.  doi: 10.4153/CJM-1979-101-3.
      I. Kombe , On the nonexistence of positive solutions to nonlinear degenerate parabolic equations with singular coefficients, Appl. Anal., 85 (2006) , 467-478.  doi: 10.1080/00036810500404967.
      B. Lian , Some sharp Rellich type inequalities on nilpotent groups and application, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013) , 59-74.  doi: 10.1016/S0252-9602(12)60194-5.
      P. Lindqvist , On the equation $\text{div}(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2}u = 0$, Proc. Amer. Math. Soc., 109 (1990) , 157-164.  doi: 10.2307/2048375.
      G. Metafune , M. Sobajima  and  S. C. Motohiro , Weighted Calderón-Zygmund and Rellich inequalities in Lp, Math. Ann., 361 (2015) , 313-366.  doi: 10.1007/s00208-014-1075-x.
      A. Moradifam , Optimal weighted Hardy-Rellich inequalities on $H^{2}\cap H_{0}^{1}$, J. Lond. Math. Soc., 85 (2012) , 22-40.  doi: 10.1112/jlms/jdr045.
      R. Musina , Weighted Sobolev spaces of radially symmetric functions, Annali di Matematica, 193 (2014) , 1629-1659.  doi: 10.1007/s10231-013-0348-4.
      P. Niu , Y. Ou  and  J. Han , Several Hardy type inequalities with weights related to generalized Greiner operator, Canad. Math. Bull., 53 (2010) , 153-162.  doi: 10.4153/CMB-2010-029-9.
      P. Niu , H. Zhang  and  Y. Wang , Hardy type and Rellich type inequalities on the Heisenberg group, Proc. Amer. Math. Soc., 129 (2001) , 3623-3630.  doi: 10.1090/S0002-9939-01-06011-7.
      F. Rellich, Halbbeschränkte Differentialoperatoren höherer Ordnung, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. Ⅲ, Erven P. Noordhoff N.V., Groningen; North-Holland Publishing Co., Amsterdam, 1956, pp. 243–250.
      A. Tertikas  and  N. Zographopoulos , Best constants in the Hardy-Rellich inequalities and related improvements, Adv. in Math., 209 (2007) , 407-459.  doi: 10.1016/j.aim.2006.05.011.
      A. Yener, General weighted Hardy type inequalities related to Greiner operators, to appear in Rocky Mountain J. Math., https://projecteuclid.org/euclid.rmjm/1528164034.
  • 加载中

Article Metrics

HTML views(342) PDF downloads(323) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint