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April  2017, 37(4): 2009-2021. doi: 10.3934/dcds.2017085

A unified approach to weighted Hardy type inequalities on Carnot groups

Jerome A. Goldstein 1, , Ismail Kombe 2, and  Abdullah Yener 2,

1. 

Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA
2. 

Department of Mathematics, Faculty of Art and Science, Istanbul Commerce University, Beyoglu 34445, Istanbul, Turkey

Received  September 2016 Revised  October 2016 Published  December 2016

We find a simple sufficient criterion on a pair of nonnegative weight functions
$V(x)$
 and
$W(x) $
 on a Carnot group
$\mathbb{G},$
 so that the general weighted
$L^{p}$
 Hardy type inequality
$\begin{equation*}\int_{\mathbb{G}}V\left( x\right) \left\vert \nabla _{\mathbb{G}}\phi \left(x\right) \right\vert ^{p}dx\geq \int_{\mathbb{G}}W\left( x\right) \left\vert\phi \left( x\right) \right\vert ^{p}dx\end{equation*}$
is valid for any
$φ ∈ C_{0}^{∞ }(\mathbb{G})$
 and
$p>1.$
 It is worth noting here that our unifying method may be readily used both to recover most of the previously known weighted Hardy and Heisenberg-Pauli-Weyl type inequalities as well as to construct other new inequalities with an explicit best constant on
$\mathbb{G}.$
 We also present some new results on two-weight
$L^{p}$
 Hardy type inequalities with remainder terms on a bounded domain
$Ω$
 in
$\mathbb{G}$
 via a differential inequality.
Keywords: Carnot groups, weighted Hardy inequality, Heisenberg-Pauli-Weyl inequality, two-weight Hardy inequality.
Mathematics Subject Classification: Primary:26D10, 22E30;Secondary:43A80.
Citation: Jerome A. Goldstein, Ismail Kombe, Abdullah Yener. A unified approach to weighted Hardy type inequalities on Carnot groups. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2009-2021. doi: 10.3934/dcds.2017085
References:
[1]

Adimurthi, M. Ramaswamy and N. Chaudhuri, An improved Hardy Sobolev inequality and its applications, Proc. Amer. Math. Soc., 130 (2002), 489-505.  doi: 10.1090/S0002-9939-01-06132-9.

[2]

Adimurthi and K. Sandeep, Existence and non-existence of the first eigenvalue of the perturbed Hardy-Sobolev operator, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1021-1043.  doi: 10.1017/S0308210500001992.

[3]

Z. Balogh and J. Tyson, Polar coordinates in Carnot groups, Math. Z., 241 (2002), 697-730.  doi: 10.1007/s00209-002-0441-7.

[4]

Z. Balogh, I. Holopainen and J. Tyson, Singular solutions, homogeneous norms and quasiconformal mappings in Carnot groups, Math. Ann., 324 (2002), 159-186.  doi: 10.1007/s00208-002-0334-4.

[5]

P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139.  doi: 10.1090/S0002-9947-1984-0742415-3.

[6]

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

[7]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for their Sub-Laplacians Springer-Verlag, Berlin-Heidelberg, 2007.

[8]

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

[9]

C. Cowan, Optimal Hardy inequalities for general elliptic operators with improvements, Commun. Pure Appl. Anal., 9 (2010), 109-140.  doi: 10.3934/cpaa.2010.9.109.

[10]

D. Danielli, N. Garofalo and N. C. Phuc, Hardy-Sobolev type inequalities with sharp constants in Carnot-Carath éodory spaces, Potential Anal., 34 (2011), 223-242.  doi: 10.1007/s11118-010-9190-0.

[11]

L. D'Ambrosio, Hardy type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci., 5 (2005), 451-486. 

[12]

D. E. Edmunds and H. Triebel, Sharp Sobolev embeddings and related Hardy inequalities: the critical case, Math. Nachr., 207 (1999), 79-92.  doi: 10.1002/mana.1999.3212070105.

[13]

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

[14]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv für Math., 13 (1975), 161-207.  doi: 10.1007/BF02386204.

[15]

G. B. Folland and E. Stein, Hardy Spaces on Homogeneous Groups Princeton University Press, Princeton, NJ, 1982.

[16]

G. B. Folland and A. Sitaram, The uncertainty principle: A mathematical survey, J. Fourier Anal. Appl., 3 (1997), 207-238.  doi: 10.1007/BF02649110.

[17]

B. Franchi, R. Serapioni and F. Serra Cassano, Regular hypersurfaces, intrinsic perimeter and implicit function theorem in carnot groups, Comm. Anal. Geom., 11 (2003), 909-944.  doi: 10.4310/CAG.2003.v11.n5.a4.

[18]

J. Garcia Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Diff. Equations, 144 (1998), 441-476.  doi: 10.1006/jdeq.1997.3375.

[19]

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

[20]

G. R. Goldstein, J. A. Goldstein and A. Rhandi, Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential, Appl. Anal., 91 (2011), 2057-2071.  doi: 10.1080/00036811.2011.587809.

[21]

J. A. Goldstein, D. Hauer and A. Rhandi, Existence and nonexistence of positive solutions of p-Kolmogorov equations perturbed by a Hardy potential, Nonlinear Anal., 131 (2016), 121-154.  doi: 10.1016/j.na.2015.07.016.

[22]

J. A. Goldstein and I. Kombe, The Hardy inequality and nonlinear parabolic equations on Carnot groups, Nonlinear Anal., 69 (2008), 4643-4653.  doi: 10.1016/j.na.2007.11.020.

[23]

D. Hauer and A. Rhandi, A weighted Hardy inequality and nonexistence of positive solutions, Arch. Math.(Basel), 100 (2013), 273-287.  doi: 10.1007/s00013-013-0484-5.

[24]

Y. Han and P. Niu, Some Hardy type inequalities in the Heisenberg group, J. Inequal. Pure Appl. Math. , 4 (2003), Article 103, 5 pp.

[25]

W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Original Scientific Papers Wissenschaftliche Originalarbeiten, A/1 (1985), 478-504.  doi: 10.1007/978-3-642-61659-4_30.

[26]

Y. Jin and S. Shen, Weighted Hardy and Rellich inequality on Carnot groups, Arch. Math.(Basel), 96 (2011), 263-271.  doi: 10.1007/s00013-011-0220-y.

[27]

I. Kombe, Sharp Weighted Rellich and uncertainty principle inequalities on Carnot groups, Comm. App. Anal., 14 (2010), 251-271. 

[28]

I. Kombe and A. Yener, Weighted Hardy and Rellich type inequalities on Riemannian manifolds, Math. Nachr., 289 (2016), 994-1004.  doi: 10.1002/mana.201500237.

[29]

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.

[30]

P. Lindqvist, On the equation $\text{div}(|\nabla u|^{p-2}\nabla u)+λ |u|^{p-2}u=0$, Proc. Amer. Math. Soc., 109 (1990), 157-164.  doi: 10.1090/S0002-9939-1990-1007505-7.

[31]

I. Skrzypczak, Hardy type inequalities derived from $p$-harmonic problems, Nonlinear Anal., 93 (2013), 30-50.  doi: 10.1016/j.na.2013.07.006.

[32]

E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals Princeton University Press, Princeton, NJ, 1993.

[33]

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.

[34]

J. Wang and P. Niu, Sharp weighted Hardy type inequalities and Hardy-Sobolev type inequalities on polarizable Carnot groups, C. R. Math. Acad. Sci. Paris Ser. I, 346 (2008), 1231-1234.  doi: 10.1016/j.crma.2008.10.009.

[35]

H. Weyl, The Theory of Groups and Quantum Mechanics Reprint of the 1931 English translation. Dover Publications, Inc. , New York, 1950.

[36]

A. Yener, Weighted Hardy type inequalities on the Heisenberg group $\mathbb{H}^{n}$, Math. Inequal. Appl., 19 (2016), 671-683.  doi: 10.7153/mia-19-48.

show all references

References:
[1]

Adimurthi, M. Ramaswamy and N. Chaudhuri, An improved Hardy Sobolev inequality and its applications, Proc. Amer. Math. Soc., 130 (2002), 489-505.  doi: 10.1090/S0002-9939-01-06132-9.

[2]

Adimurthi and K. Sandeep, Existence and non-existence of the first eigenvalue of the perturbed Hardy-Sobolev operator, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1021-1043.  doi: 10.1017/S0308210500001992.

[3]

Z. Balogh and J. Tyson, Polar coordinates in Carnot groups, Math. Z., 241 (2002), 697-730.  doi: 10.1007/s00209-002-0441-7.

[4]

Z. Balogh, I. Holopainen and J. Tyson, Singular solutions, homogeneous norms and quasiconformal mappings in Carnot groups, Math. Ann., 324 (2002), 159-186.  doi: 10.1007/s00208-002-0334-4.

[5]

P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139.  doi: 10.1090/S0002-9947-1984-0742415-3.

[6]

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

[7]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for their Sub-Laplacians Springer-Verlag, Berlin-Heidelberg, 2007.

[8]

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

[9]

C. Cowan, Optimal Hardy inequalities for general elliptic operators with improvements, Commun. Pure Appl. Anal., 9 (2010), 109-140.  doi: 10.3934/cpaa.2010.9.109.

[10]

D. Danielli, N. Garofalo and N. C. Phuc, Hardy-Sobolev type inequalities with sharp constants in Carnot-Carath éodory spaces, Potential Anal., 34 (2011), 223-242.  doi: 10.1007/s11118-010-9190-0.

[11]

L. D'Ambrosio, Hardy type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci., 5 (2005), 451-486. 

[12]

D. E. Edmunds and H. Triebel, Sharp Sobolev embeddings and related Hardy inequalities: the critical case, Math. Nachr., 207 (1999), 79-92.  doi: 10.1002/mana.1999.3212070105.

[13]

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

[14]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv für Math., 13 (1975), 161-207.  doi: 10.1007/BF02386204.

[15]

G. B. Folland and E. Stein, Hardy Spaces on Homogeneous Groups Princeton University Press, Princeton, NJ, 1982.

[16]

G. B. Folland and A. Sitaram, The uncertainty principle: A mathematical survey, J. Fourier Anal. Appl., 3 (1997), 207-238.  doi: 10.1007/BF02649110.

[17]

B. Franchi, R. Serapioni and F. Serra Cassano, Regular hypersurfaces, intrinsic perimeter and implicit function theorem in carnot groups, Comm. Anal. Geom., 11 (2003), 909-944.  doi: 10.4310/CAG.2003.v11.n5.a4.

[18]

J. Garcia Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Diff. Equations, 144 (1998), 441-476.  doi: 10.1006/jdeq.1997.3375.

[19]

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

[20]

G. R. Goldstein, J. A. Goldstein and A. Rhandi, Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential, Appl. Anal., 91 (2011), 2057-2071.  doi: 10.1080/00036811.2011.587809.

[21]

J. A. Goldstein, D. Hauer and A. Rhandi, Existence and nonexistence of positive solutions of p-Kolmogorov equations perturbed by a Hardy potential, Nonlinear Anal., 131 (2016), 121-154.  doi: 10.1016/j.na.2015.07.016.

[22]

J. A. Goldstein and I. Kombe, The Hardy inequality and nonlinear parabolic equations on Carnot groups, Nonlinear Anal., 69 (2008), 4643-4653.  doi: 10.1016/j.na.2007.11.020.

[23]

D. Hauer and A. Rhandi, A weighted Hardy inequality and nonexistence of positive solutions, Arch. Math.(Basel), 100 (2013), 273-287.  doi: 10.1007/s00013-013-0484-5.

[24]

Y. Han and P. Niu, Some Hardy type inequalities in the Heisenberg group, J. Inequal. Pure Appl. Math. , 4 (2003), Article 103, 5 pp.

[25]

W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Original Scientific Papers Wissenschaftliche Originalarbeiten, A/1 (1985), 478-504.  doi: 10.1007/978-3-642-61659-4_30.

[26]

Y. Jin and S. Shen, Weighted Hardy and Rellich inequality on Carnot groups, Arch. Math.(Basel), 96 (2011), 263-271.  doi: 10.1007/s00013-011-0220-y.

[27]

I. Kombe, Sharp Weighted Rellich and uncertainty principle inequalities on Carnot groups, Comm. App. Anal., 14 (2010), 251-271. 

[28]

I. Kombe and A. Yener, Weighted Hardy and Rellich type inequalities on Riemannian manifolds, Math. Nachr., 289 (2016), 994-1004.  doi: 10.1002/mana.201500237.

[29]

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.

[30]

P. Lindqvist, On the equation $\text{div}(|\nabla u|^{p-2}\nabla u)+λ |u|^{p-2}u=0$, Proc. Amer. Math. Soc., 109 (1990), 157-164.  doi: 10.1090/S0002-9939-1990-1007505-7.

[31]

I. Skrzypczak, Hardy type inequalities derived from $p$-harmonic problems, Nonlinear Anal., 93 (2013), 30-50.  doi: 10.1016/j.na.2013.07.006.

[32]

E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals Princeton University Press, Princeton, NJ, 1993.

[33]

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.

[34]

J. Wang and P. Niu, Sharp weighted Hardy type inequalities and Hardy-Sobolev type inequalities on polarizable Carnot groups, C. R. Math. Acad. Sci. Paris Ser. I, 346 (2008), 1231-1234.  doi: 10.1016/j.crma.2008.10.009.

[35]

H. Weyl, The Theory of Groups and Quantum Mechanics Reprint of the 1931 English translation. Dover Publications, Inc. , New York, 1950.

[36]

A. Yener, Weighted Hardy type inequalities on the Heisenberg group $\mathbb{H}^{n}$, Math. Inequal. Appl., 19 (2016), 671-683.  doi: 10.7153/mia-19-48.

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