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 and Applied Analysis, 2016, 15 (5) : 1871-1892. doi: 10.3934/cpaa.2016020
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, vol. 29, Princeton University Press, Princeton, 1982.

[2]

K. Akutagawa, H. Kumura, Geometric relative Hardy inequalities and the discrete spectrum of Schrödinger operators on manifolds, Calc. Var. Part. Diff. Eq., 48 (2013), 67-88. doi: 10.1007/s00526-012-0542-z.

[3]

E. Berchio, D. Ganguly and G. Grillo, Sharp Poincaré-Hardy and Poincaré-Rellich inequalities on the hyperbolic space, Preprint 2015, arXiv:1507.02550v2.

[4]

G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants, Trans. Amer. Soc, 356 (2004), 2169-2196. doi: 10.1090/S0002-9947-03-03389-0.

[5]

G. Barbatis, S. Filippas and A. Tertikas, Series expansion for $L^p$ Hardy inequalities, Indiana Univ. Math. J., 52 (2003), 171-190. doi: 10.1512/iumj.2003.52.2207.

[6]

G. Barbatis and A. Tertikas, On a class of Rellich inequalities, J. Comput. Appl. Math., 194 (2006), 156-172. doi: 10.1016/j.cam.2005.06.020.

[7]

B. Bianchini, L. Mari and M. Rigoli, Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem, J. Funct. Anal., 268 (2015), 1-72. doi: 10.1016/j.jfa.2014.10.016.

[8]

Y. Bozhkov and E. Mitidieri, Conformal Killing vector fields and Rellich type identities on Riemannian manifolds, I Lecture Notes of Seminario Interdisciplinare di Matematica, 7 (2008), 65-80.

[9]

Y. Bozhkov and E. Mitidieri, Conformal Killing vector fields and Rellich type identities on Riemannian manifolds II, Mediterr. J. Math., 9 (2012), 1-20.

[10]

H. Brezis and M. Marcus, Hardy's inequalities revisited, Ann. Scuola Norm. Sup. Cl. Sci., 25 (1997), 217-237.

[11]

H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.

[12]

G. Carron, Inegalites de Hardy sur les varietes Riemanniennes non-compactes, J. Math. Pures Appl., 76 (1997), 883-891. doi: 10.1016/S0021-7824(97)89976-X.

[13]

L. D'Ambrosio and S. Dipierro, Hardy inequalities on Riemannian manifolds and applications, Ann. Inst. H. Poinc. Anal. Non Lin., 31 (2014), 449-475. doi: 10.1016/j.anihpc.2013.04.004.

[14]

E. B. Davies and A. M. Hinz, Explicit constants for Rellich inequalities in $L^p(\Omega)$, Math. Z., 227 (1998), 511-523. doi: 10.1007/PL00004389.

[15]

B. Devyver, M. Fraas and Y. Pinchover, Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon, J. Funct. Anal., 266 (2014), 4422-4489. doi: 10.1016/j.jfa.2014.01.017.

[16]

S. Filippas, L. Moschini and A. Tertikas, Sharp trace Hardy-Sobolev-Maz'ya inequalities and the fractional Laplacian, Arch. Ration. Mech. Anal., 208 (2013), 109-161. doi: 10.1007/s00205-012-0594-4.

[17]

S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), 186-233. doi: 10.1006/jfan.2001.3900.

[18]

S. Filippas, A. Tertikas and J. Tidblom, On the structure of Hardy-Sobolev-Maz'ya inequalities, J. Eur. Math. Soc., 11 (2009), 1165-1185. doi: 10.4171/JEMS/178.

[19]

F. Gazzola, H. 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.

[20]

N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349 (2011), 1-57. doi: 10.1007/s00208-010-0510-x.

[21]

I. Kombe and M. Ozaydin, Improved Hardy and Rellich inequalities on Riemannian manifolds, Trans. Amer. Math. Soc., 361 (2009), 6191-6203. doi: 10.1090/S0002-9947-09-04642-X.

[22]

I. Kombe and M. Ozaydin, Rellich and uncertainty principle inequalities on Riemannian manifolds, Trans. Amer. Math. Soc., 365 (2013), 5035-5050. doi: 10.1090/S0002-9947-2013-05763-7.

[23]

D. Karmakar and K. Sandeep, Adams Inequality on the Hyperbolic space, J. Funct. Anal., 270 (2016), 1792-1817. doi: 10.1016/j.jfa.2015.11.019.

[24]

P. Li and J. Wang, Weighted Poincaré inequality and rigidity of complete manifolds, Ann. Sci. École Norm. Sup., 39 (2006), 921-982. doi: 10.1016/j.ansens.2006.11.001.

[25]

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

[26]

G. Mancini and K. Sandeep, On a semilinear equation in $\mathbbH^n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 vol.VII (2008), 635-671.

[27]

M. Marcus, V. J. Mizel and Y. Pinchover, On the best constant for Hardy's inequality in $\mathbb R^n$, Trans. Am. Math. Soc., 350 (1998), 3237-3255. doi: 10.1090/S0002-9947-98-02122-9.

[28]

G. Metafune, M. Sobajima and C. Spina, Weighted Calderón-ygmund and Rellich inequalities in $L^p$, Math. Ann., 361 (2015), 313-366. doi: 10.1007/s00208-014-1075-x.

[29]

E. Mitidieri, A simple approach to Hardy inequalities, Mat. Zametki, 67 (2000), 563-572. doi: 10.1007/BF02676404.

[30]

F. Rellich, Halbbeschrkte differential operatoren herer Ordnung, Proceedings of the International Congress of Mathematicians III (1954), 243-250.

[31]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 32, Princeton (1971).

[32]

A. Tertikas and N. B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements, Adv. Math., 209 (2007), 407-459. doi: 10.1016/j.aim.2006.05.011.

[33]

Q. Yang, D. Su and Y. Kong, Hardy inequalities on Riemannian manifolds with negative curvature, Commun. Contemp. Math., 16 (2014), 1350043. doi: 10.1142/S0219199713500430.

[34]

J. L. Vazquez, Fundamental solution and long time behaviour of the Porous medium equation in hyperbolic space, J. Math. Pures Appl., 104 (2015), 454-484. doi: 10.1016/j.matpur.2015.03.005.

[35]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556.

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, vol. 29, Princeton University Press, Princeton, 1982.

[2]

K. Akutagawa, H. Kumura, Geometric relative Hardy inequalities and the discrete spectrum of Schrödinger operators on manifolds, Calc. Var. Part. Diff. Eq., 48 (2013), 67-88. doi: 10.1007/s00526-012-0542-z.

[3]

E. Berchio, D. Ganguly and G. Grillo, Sharp Poincaré-Hardy and Poincaré-Rellich inequalities on the hyperbolic space, Preprint 2015, arXiv:1507.02550v2.

[4]

G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants, Trans. Amer. Soc, 356 (2004), 2169-2196. doi: 10.1090/S0002-9947-03-03389-0.

[5]

G. Barbatis, S. Filippas and A. Tertikas, Series expansion for $L^p$ Hardy inequalities, Indiana Univ. Math. J., 52 (2003), 171-190. doi: 10.1512/iumj.2003.52.2207.

[6]

G. Barbatis and A. Tertikas, On a class of Rellich inequalities, J. Comput. Appl. Math., 194 (2006), 156-172. doi: 10.1016/j.cam.2005.06.020.

[7]

B. Bianchini, L. Mari and M. Rigoli, Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem, J. Funct. Anal., 268 (2015), 1-72. doi: 10.1016/j.jfa.2014.10.016.

[8]

Y. Bozhkov and E. Mitidieri, Conformal Killing vector fields and Rellich type identities on Riemannian manifolds, I Lecture Notes of Seminario Interdisciplinare di Matematica, 7 (2008), 65-80.

[9]

Y. Bozhkov and E. Mitidieri, Conformal Killing vector fields and Rellich type identities on Riemannian manifolds II, Mediterr. J. Math., 9 (2012), 1-20.

[10]

H. Brezis and M. Marcus, Hardy's inequalities revisited, Ann. Scuola Norm. Sup. Cl. Sci., 25 (1997), 217-237.

[11]

H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.

[12]

G. Carron, Inegalites de Hardy sur les varietes Riemanniennes non-compactes, J. Math. Pures Appl., 76 (1997), 883-891. doi: 10.1016/S0021-7824(97)89976-X.

[13]

L. D'Ambrosio and S. Dipierro, Hardy inequalities on Riemannian manifolds and applications, Ann. Inst. H. Poinc. Anal. Non Lin., 31 (2014), 449-475. doi: 10.1016/j.anihpc.2013.04.004.

[14]

E. B. Davies and A. M. Hinz, Explicit constants for Rellich inequalities in $L^p(\Omega)$, Math. Z., 227 (1998), 511-523. doi: 10.1007/PL00004389.

[15]

B. Devyver, M. Fraas and Y. Pinchover, Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon, J. Funct. Anal., 266 (2014), 4422-4489. doi: 10.1016/j.jfa.2014.01.017.

[16]

S. Filippas, L. Moschini and A. Tertikas, Sharp trace Hardy-Sobolev-Maz'ya inequalities and the fractional Laplacian, Arch. Ration. Mech. Anal., 208 (2013), 109-161. doi: 10.1007/s00205-012-0594-4.

[17]

S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), 186-233. doi: 10.1006/jfan.2001.3900.

[18]

S. Filippas, A. Tertikas and J. Tidblom, On the structure of Hardy-Sobolev-Maz'ya inequalities, J. Eur. Math. Soc., 11 (2009), 1165-1185. doi: 10.4171/JEMS/178.

[19]

F. Gazzola, H. 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.

[20]

N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349 (2011), 1-57. doi: 10.1007/s00208-010-0510-x.

[21]

I. Kombe and M. Ozaydin, Improved Hardy and Rellich inequalities on Riemannian manifolds, Trans. Amer. Math. Soc., 361 (2009), 6191-6203. doi: 10.1090/S0002-9947-09-04642-X.

[22]

I. Kombe and M. Ozaydin, Rellich and uncertainty principle inequalities on Riemannian manifolds, Trans. Amer. Math. Soc., 365 (2013), 5035-5050. doi: 10.1090/S0002-9947-2013-05763-7.

[23]

D. Karmakar and K. Sandeep, Adams Inequality on the Hyperbolic space, J. Funct. Anal., 270 (2016), 1792-1817. doi: 10.1016/j.jfa.2015.11.019.

[24]

P. Li and J. Wang, Weighted Poincaré inequality and rigidity of complete manifolds, Ann. Sci. École Norm. Sup., 39 (2006), 921-982. doi: 10.1016/j.ansens.2006.11.001.

[25]

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

[26]

G. Mancini and K. Sandeep, On a semilinear equation in $\mathbbH^n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 vol.VII (2008), 635-671.

[27]

M. Marcus, V. J. Mizel and Y. Pinchover, On the best constant for Hardy's inequality in $\mathbb R^n$, Trans. Am. Math. Soc., 350 (1998), 3237-3255. doi: 10.1090/S0002-9947-98-02122-9.

[28]

G. Metafune, M. Sobajima and C. Spina, Weighted Calderón-ygmund and Rellich inequalities in $L^p$, Math. Ann., 361 (2015), 313-366. doi: 10.1007/s00208-014-1075-x.

[29]

E. Mitidieri, A simple approach to Hardy inequalities, Mat. Zametki, 67 (2000), 563-572. doi: 10.1007/BF02676404.

[30]

F. Rellich, Halbbeschrkte differential operatoren herer Ordnung, Proceedings of the International Congress of Mathematicians III (1954), 243-250.

[31]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 32, Princeton (1971).

[32]

A. Tertikas and N. B. Zographopoulos, Best constants in the Hardy-Rellich inequalities and related improvements, Adv. Math., 209 (2007), 407-459. doi: 10.1016/j.aim.2006.05.011.

[33]

Q. Yang, D. Su and Y. Kong, Hardy inequalities on Riemannian manifolds with negative curvature, Commun. Contemp. Math., 16 (2014), 1350043. doi: 10.1142/S0219199713500430.

[34]

J. L. Vazquez, Fundamental solution and long time behaviour of the Porous medium equation in hyperbolic space, J. Math. Pures Appl., 104 (2015), 454-484. doi: 10.1016/j.matpur.2015.03.005.

[35]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556.

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