September  2016, 15(5): 1871-1892. doi: 10.3934/cpaa.2016020

Improved higher order poincaré inequalities on the hyperbolic space via Hardy-type remainder terms

1. 

Dipartimento SEMEQ, Università del Piemonte Orientale, via E. Perrone 18, Novara, 28100

2. 

Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel

Received  November 2015 Revised  April 2016 Published  July 2016

The paper deals about Hardy-type inequalities associated with the following higher order Poincaré inequality: $$ \left( \frac{N-1}{2} \right)^{2(k -l)} :=\displaystyle \inf_{ u \in C^{\infty}_{0}(\mathbb{H}^{N} ) \setminus \{0\}} \qquad \frac{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^2 \ dv_{\mathbb{H}^{N}}}{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{l} u|^2 \ dv_{\mathbb{H}^{N}} }, $$ where $0 \leq l < k$ are integers and $\mathbb{H}^{N}$ denotes the hyperbolic space. More precisely, we improve the Poincaré inequality associated with the above ratio by showing the existence of $k$ Hardy-type remainder terms. Furthermore, when $k = 2$ and $l = 1$ the existence of further remainder terms are provided and the sharpness of some constants is also discussed. As an application, we derive improved Rellich type inequalities on upper half space of the Euclidean space with non-standard remainder terms.
Citation: Elvise Berchio, Debdip Ganguly. Improved higher order poincaré inequalities on the hyperbolic space via Hardy-type remainder terms. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1871-1892. doi: 10.3934/cpaa.2016020
References:
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S. Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-body Schröinger Operators,, Math. Notes, (1982). Google Scholar

[2]

K. Akutagawa, H. Kumura, Geometric relative Hardy inequalities and the discrete spectrum of Schrödinger operators on manifolds,, \emph{Calc. Var. Part. Diff. Eq.}, 48 (2013), 67. doi: 10.1007/s00526-012-0542-z. Google Scholar

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E. Berchio, D. Ganguly and G. Grillo, Sharp Poincaré-Hardy and Poincaré-Rellich inequalities on the hyperbolic space,, Preprint 2015, (2015). Google Scholar

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G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants,, \emph{Trans. Amer. Soc}, 356 (2004), 2169. doi: 10.1090/S0002-9947-03-03389-0. Google Scholar

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G. Barbatis, S. Filippas and A. Tertikas, Series expansion for $L^p$ Hardy inequalities,, \emph{Indiana Univ. Math. J.}, 52 (2003), 171. doi: 10.1512/iumj.2003.52.2207. Google Scholar

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G. Barbatis and A. Tertikas, On a class of Rellich inequalities,, \emph{J. Comput. Appl. Math.}, 194 (2006), 156. doi: 10.1016/j.cam.2005.06.020. Google Scholar

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B. Bianchini, L. Mari and M. Rigoli, Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem,, \emph{J. Funct. Anal.}, 268 (2015), 1. doi: 10.1016/j.jfa.2014.10.016. Google Scholar

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Y. Bozhkov and E. Mitidieri, Conformal Killing vector fields and Rellich type identities on Riemannian manifolds,, \emph{I Lecture Notes of Seminario Interdisciplinare di Matematica}, 7 (2008), 65. Google Scholar

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Y. Bozhkov and E. Mitidieri, Conformal Killing vector fields and Rellich type identities on Riemannian manifolds II,, \emph{Mediterr. J. Math.}, 9 (2012), 1. Google Scholar

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H. Brezis and M. Marcus, Hardy's inequalities revisited,, \emph{Ann. Scuola Norm. Sup. Cl. Sci.}, 25 (1997), 217. Google Scholar

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H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems,, \emph{Rev. Mat. Univ. Complut. Madrid}, 10 (1997), 443. Google Scholar

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G. Carron, Inegalites de Hardy sur les varietes Riemanniennes non-compactes,, \emph{J. Math. Pures Appl.}, 76 (1997), 883. doi: 10.1016/S0021-7824(97)89976-X. Google Scholar

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L. D'Ambrosio and S. Dipierro, Hardy inequalities on Riemannian manifolds and applications,, \emph{Ann. Inst. H. Poinc. Anal. Non Lin.}, 31 (2014), 449. doi: 10.1016/j.anihpc.2013.04.004. Google Scholar

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B. Devyver, M. Fraas and Y. Pinchover, Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon,, \emph{J. Funct. Anal.}, 266 (2014), 4422. doi: 10.1016/j.jfa.2014.01.017. Google Scholar

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S. Filippas, L. Moschini and A. Tertikas, Sharp trace Hardy-Sobolev-Maz'ya inequalities and the fractional Laplacian,, \emph{Arch. Ration. Mech. Anal.}, 208 (2013), 109. doi: 10.1007/s00205-012-0594-4. Google Scholar

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S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities,, \emph{J. Funct. Anal.}, 192 (2002), 186. doi: 10.1006/jfan.2001.3900. Google Scholar

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S. Filippas, A. Tertikas and J. Tidblom, On the structure of Hardy-Sobolev-Maz'ya inequalities,, \emph{J. Eur. Math. Soc.}, 11 (2009), 1165. doi: 10.4171/JEMS/178. Google Scholar

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F. Gazzola, H. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms,, \emph{Trans. Amer. Math. Soc.}, 356 (2004), 2149. doi: 10.1090/S0002-9947-03-03395-6. Google Scholar

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N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities,, \emph{Math. Ann.}, 349 (2011), 1. doi: 10.1007/s00208-010-0510-x. Google Scholar

[21]

I. Kombe and M. Ozaydin, Improved Hardy and Rellich inequalities on Riemannian manifolds,, \emph{Trans. Amer. Math. Soc.}, 361 (2009), 6191. doi: 10.1090/S0002-9947-09-04642-X. Google Scholar

[22]

I. Kombe and M. Ozaydin, Rellich and uncertainty principle inequalities on Riemannian manifolds,, \emph{Trans. Amer. Math. Soc.}, 365 (2013), 5035. doi: 10.1090/S0002-9947-2013-05763-7. Google Scholar

[23]

D. Karmakar and K. Sandeep, Adams Inequality on the Hyperbolic space,, \emph{J. Funct. Anal.}, 270 (2016), 1792. doi: 10.1016/j.jfa.2015.11.019. Google Scholar

[24]

P. Li and J. Wang, Weighted Poincaré inequality and rigidity of complete manifolds,, \emph{Ann. Sci. \'Ecole Norm. Sup.}, 39 (2006), 921. doi: 10.1016/j.ansens.2006.11.001. Google Scholar

[25]

V. G. Maz'ya, Sobolev Spaces,, Springer-Verlag, (1985). doi: 10.1007/978-3-662-09922-3. Google Scholar

[26]

G. Mancini and K. Sandeep, On a semilinear equation in $\mathbbH^n$,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 5 (2008), 635. Google Scholar

[27]

M. Marcus, V. J. Mizel and Y. Pinchover, On the best constant for Hardy's inequality in $\mathbb R^n$,, \emph{Trans. Am. Math. Soc.}, 350 (1998), 3237. doi: 10.1090/S0002-9947-98-02122-9. Google Scholar

[28]

G. Metafune, M. Sobajima and C. Spina, Weighted Calderón-ygmund and Rellich inequalities in $L^p$,, \emph{Math. Ann.}, 361 (2015), 313. doi: 10.1007/s00208-014-1075-x. Google Scholar

[29]

E. Mitidieri, A simple approach to Hardy inequalities,, \emph{Mat. Zametki}, 67 (2000), 563. doi: 10.1007/BF02676404. Google Scholar

[30]

F. Rellich, Halbbeschrkte differential operatoren herer Ordnung,, \emph{Proceedings of the International Congress of Mathematicians III} (1954), (1954), 243. Google Scholar

[31]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces,, Princeton University Press, (1971). Google Scholar

[32]

A. Tertikas and N. B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements,, \emph{Adv. Math.}, 209 (2007), 407. doi: 10.1016/j.aim.2006.05.011. Google Scholar

[33]

Q. Yang, D. Su and Y. Kong, Hardy inequalities on Riemannian manifolds with negative curvature,, \emph{Commun. Contemp. Math.}, 16 (2014). doi: 10.1142/S0219199713500430. Google Scholar

[34]

J. L. Vazquez, Fundamental solution and long time behaviour of the Porous medium equation in hyperbolic space,, \emph{J. Math. Pures Appl.}, 104 (2015), 454. doi: 10.1016/j.matpur.2015.03.005. Google Scholar

[35]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential,, \emph{J. Funct. Anal.}, 173 (2000), 103. doi: 10.1006/jfan.1999.3556. Google Scholar

show all references

References:
[1]

S. Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-body Schröinger Operators,, Math. Notes, (1982). Google Scholar

[2]

K. Akutagawa, H. Kumura, Geometric relative Hardy inequalities and the discrete spectrum of Schrödinger operators on manifolds,, \emph{Calc. Var. Part. Diff. Eq.}, 48 (2013), 67. doi: 10.1007/s00526-012-0542-z. Google Scholar

[3]

E. Berchio, D. Ganguly and G. Grillo, Sharp Poincaré-Hardy and Poincaré-Rellich inequalities on the hyperbolic space,, Preprint 2015, (2015). Google Scholar

[4]

G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants,, \emph{Trans. Amer. Soc}, 356 (2004), 2169. doi: 10.1090/S0002-9947-03-03389-0. Google Scholar

[5]

G. Barbatis, S. Filippas and A. Tertikas, Series expansion for $L^p$ Hardy inequalities,, \emph{Indiana Univ. Math. J.}, 52 (2003), 171. doi: 10.1512/iumj.2003.52.2207. Google Scholar

[6]

G. Barbatis and A. Tertikas, On a class of Rellich inequalities,, \emph{J. Comput. Appl. Math.}, 194 (2006), 156. doi: 10.1016/j.cam.2005.06.020. Google Scholar

[7]

B. Bianchini, L. Mari and M. Rigoli, Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem,, \emph{J. Funct. Anal.}, 268 (2015), 1. doi: 10.1016/j.jfa.2014.10.016. Google Scholar

[8]

Y. Bozhkov and E. Mitidieri, Conformal Killing vector fields and Rellich type identities on Riemannian manifolds,, \emph{I Lecture Notes of Seminario Interdisciplinare di Matematica}, 7 (2008), 65. Google Scholar

[9]

Y. Bozhkov and E. Mitidieri, Conformal Killing vector fields and Rellich type identities on Riemannian manifolds II,, \emph{Mediterr. J. Math.}, 9 (2012), 1. Google Scholar

[10]

H. Brezis and M. Marcus, Hardy's inequalities revisited,, \emph{Ann. Scuola Norm. Sup. Cl. Sci.}, 25 (1997), 217. Google Scholar

[11]

H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems,, \emph{Rev. Mat. Univ. Complut. Madrid}, 10 (1997), 443. Google Scholar

[12]

G. Carron, Inegalites de Hardy sur les varietes Riemanniennes non-compactes,, \emph{J. Math. Pures Appl.}, 76 (1997), 883. doi: 10.1016/S0021-7824(97)89976-X. Google Scholar

[13]

L. D'Ambrosio and S. Dipierro, Hardy inequalities on Riemannian manifolds and applications,, \emph{Ann. Inst. H. Poinc. Anal. Non Lin.}, 31 (2014), 449. doi: 10.1016/j.anihpc.2013.04.004. Google Scholar

[14]

E. B. Davies and A. M. Hinz, Explicit constants for Rellich inequalities in $L^p(\Omega)$,, \emph{Math. Z.}, 227 (1998), 511. doi: 10.1007/PL00004389. Google Scholar

[15]

B. Devyver, M. Fraas and Y. Pinchover, Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon,, \emph{J. Funct. Anal.}, 266 (2014), 4422. doi: 10.1016/j.jfa.2014.01.017. Google Scholar

[16]

S. Filippas, L. Moschini and A. Tertikas, Sharp trace Hardy-Sobolev-Maz'ya inequalities and the fractional Laplacian,, \emph{Arch. Ration. Mech. Anal.}, 208 (2013), 109. doi: 10.1007/s00205-012-0594-4. Google Scholar

[17]

S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities,, \emph{J. Funct. Anal.}, 192 (2002), 186. doi: 10.1006/jfan.2001.3900. Google Scholar

[18]

S. Filippas, A. Tertikas and J. Tidblom, On the structure of Hardy-Sobolev-Maz'ya inequalities,, \emph{J. Eur. Math. Soc.}, 11 (2009), 1165. doi: 10.4171/JEMS/178. Google Scholar

[19]

F. Gazzola, H. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms,, \emph{Trans. Amer. Math. Soc.}, 356 (2004), 2149. doi: 10.1090/S0002-9947-03-03395-6. Google Scholar

[20]

N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities,, \emph{Math. Ann.}, 349 (2011), 1. doi: 10.1007/s00208-010-0510-x. Google Scholar

[21]

I. Kombe and M. Ozaydin, Improved Hardy and Rellich inequalities on Riemannian manifolds,, \emph{Trans. Amer. Math. Soc.}, 361 (2009), 6191. doi: 10.1090/S0002-9947-09-04642-X. Google Scholar

[22]

I. Kombe and M. Ozaydin, Rellich and uncertainty principle inequalities on Riemannian manifolds,, \emph{Trans. Amer. Math. Soc.}, 365 (2013), 5035. doi: 10.1090/S0002-9947-2013-05763-7. Google Scholar

[23]

D. Karmakar and K. Sandeep, Adams Inequality on the Hyperbolic space,, \emph{J. Funct. Anal.}, 270 (2016), 1792. doi: 10.1016/j.jfa.2015.11.019. Google Scholar

[24]

P. Li and J. Wang, Weighted Poincaré inequality and rigidity of complete manifolds,, \emph{Ann. Sci. \'Ecole Norm. Sup.}, 39 (2006), 921. doi: 10.1016/j.ansens.2006.11.001. Google Scholar

[25]

V. G. Maz'ya, Sobolev Spaces,, Springer-Verlag, (1985). doi: 10.1007/978-3-662-09922-3. Google Scholar

[26]

G. Mancini and K. Sandeep, On a semilinear equation in $\mathbbH^n$,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 5 (2008), 635. Google Scholar

[27]

M. Marcus, V. J. Mizel and Y. Pinchover, On the best constant for Hardy's inequality in $\mathbb R^n$,, \emph{Trans. Am. Math. Soc.}, 350 (1998), 3237. doi: 10.1090/S0002-9947-98-02122-9. Google Scholar

[28]

G. Metafune, M. Sobajima and C. Spina, Weighted Calderón-ygmund and Rellich inequalities in $L^p$,, \emph{Math. Ann.}, 361 (2015), 313. doi: 10.1007/s00208-014-1075-x. Google Scholar

[29]

E. Mitidieri, A simple approach to Hardy inequalities,, \emph{Mat. Zametki}, 67 (2000), 563. doi: 10.1007/BF02676404. Google Scholar

[30]

F. Rellich, Halbbeschrkte differential operatoren herer Ordnung,, \emph{Proceedings of the International Congress of Mathematicians III} (1954), (1954), 243. Google Scholar

[31]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces,, Princeton University Press, (1971). Google Scholar

[32]

A. Tertikas and N. B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements,, \emph{Adv. Math.}, 209 (2007), 407. doi: 10.1016/j.aim.2006.05.011. Google Scholar

[33]

Q. Yang, D. Su and Y. Kong, Hardy inequalities on Riemannian manifolds with negative curvature,, \emph{Commun. Contemp. Math.}, 16 (2014). doi: 10.1142/S0219199713500430. Google Scholar

[34]

J. L. Vazquez, Fundamental solution and long time behaviour of the Porous medium equation in hyperbolic space,, \emph{J. Math. Pures Appl.}, 104 (2015), 454. doi: 10.1016/j.matpur.2015.03.005. Google Scholar

[35]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential,, \emph{J. Funct. Anal.}, 173 (2000), 103. doi: 10.1006/jfan.1999.3556. Google Scholar

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