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March  2019, 18(2): 869-886. doi: 10.3934/cpaa.2019042

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

Department of Mathematics, Faculty of Humanities and Social Sciences, Istanbul Ticaret University, Beyoglu, 34445, Istanbul, Turkey

* Corresponding author

Received  April 2018 Revised  July 2018 Published  October 2018

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
$\mathbb{R}^{2n+1}$
.
Citation: Ismail Kombe, Abdullah Yener. A general approach to weighted $L^{p}$ Rellich type inequalities related to Greiner operator. Communications on Pure & Applied Analysis, 2019, 18 (2) : 869-886. doi: 10.3934/cpaa.2019042
References:
[1]

AdimurthiM. 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.  Google Scholar

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

G. Barbatis, Improved Rellich inequalities for the polyharmonic operator, Indiana Univ. Math. J., 55 (2006), 1401-1422.  doi: 10.1512/iumj.2006.55.2752.  Google Scholar

[7]

R. BealsB. 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.  Google Scholar

[8]

R. BealsB. 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.  Google Scholar

[9]

D. M. Bennett, An extension of Rellich's inequality, Proc. Amer. Math. Soc., 106 (1989), 987-993.  doi: 10.2307/2047283.  Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

A. DetallaT. Horiuchi and H. Ando, Sharp remainder terms of the Rellich inequality and its application, Bull. Malays. Math. Sci. Soc., 35 (2012), 519-528.   Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.   Google Scholar

[16]

F. GazzolaH. 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.  Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

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.  Google Scholar

[21]

G. MetafuneM. 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.  Google Scholar

[22]

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.  Google Scholar

[23]

R. Musina, Weighted Sobolev spaces of radially symmetric functions, Annali di Matematica, 193 (2014), 1629-1659.  doi: 10.1007/s10231-013-0348-4.  Google Scholar

[24]

P. NiuY. 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.  Google Scholar

[25]

P. NiuH. 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.  Google Scholar

[26]

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.  Google Scholar

[27]

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.  Google Scholar

[28]

A. Yener, General weighted Hardy type inequalities related to Greiner operators, to appear in Rocky Mountain J. Math., https://projecteuclid.org/euclid.rmjm/1528164034. Google Scholar

show all references

References:
[1]

AdimurthiM. 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.  Google Scholar

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

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.  Google Scholar

[6]

G. Barbatis, Improved Rellich inequalities for the polyharmonic operator, Indiana Univ. Math. J., 55 (2006), 1401-1422.  doi: 10.1512/iumj.2006.55.2752.  Google Scholar

[7]

R. BealsB. 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.  Google Scholar

[8]

R. BealsB. 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.  Google Scholar

[9]

D. M. Bennett, An extension of Rellich's inequality, Proc. Amer. Math. Soc., 106 (1989), 987-993.  doi: 10.2307/2047283.  Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

A. DetallaT. Horiuchi and H. Ando, Sharp remainder terms of the Rellich inequality and its application, Bull. Malays. Math. Sci. Soc., 35 (2012), 519-528.   Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.   Google Scholar

[16]

F. GazzolaH. 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.  Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

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.  Google Scholar

[21]

G. MetafuneM. 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.  Google Scholar

[22]

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.  Google Scholar

[23]

R. Musina, Weighted Sobolev spaces of radially symmetric functions, Annali di Matematica, 193 (2014), 1629-1659.  doi: 10.1007/s10231-013-0348-4.  Google Scholar

[24]

P. NiuY. 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.  Google Scholar

[25]

P. NiuH. 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.  Google Scholar

[26]

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.  Google Scholar

[27]

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.  Google Scholar

[28]

A. Yener, General weighted Hardy type inequalities related to Greiner operators, to appear in Rocky Mountain J. Math., https://projecteuclid.org/euclid.rmjm/1528164034. Google Scholar

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