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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

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