• Previous Article
    Global existence and asymptotic behavior of spherically symmetric solutions for the multi-dimensional infrarelativistic model
  • CPAA Home
  • This Issue
  • Next Article
    Local well-posedness of the fifth-order KdV-type equations on the half-line
September 2019, 18(5): 2575-2605. doi: 10.3934/cpaa.2019116

On extension problem, trace Hardy and Hardy's inequalities for some fractional Laplacians

1. 

Department of Mathematics, BITS Pilani K K Birla Goa Campus, Zuarinagar, South Goa 403 726, Goa, India

2. 

BCAM - Basque Center for Applied Mathematics 48009 Bilbao, Spain

3. 

Ikerbasque, Basque Foundation for Science, 48011 Bilbao, Spain

4. 

Department of Mathematics, Indian Institute of Science, 560 012 Bangalore, India

* Corresponding author

Received  July 2018 Revised  January 2019 Published  April 2019

We obtain generalised trace Hardy inequalities for fractional powers of general operators given by sums of squares of vector fields. Such inequalities are derived by means of particular solutions of an extended equation associated to the above-mentioned operators. As a consequence, Hardy inequalities are also deduced. Particular cases include Laplacians on stratified groups, Euclidean motion groups and special Hermite operators. Fairly explicit expressions for the constants are provided. Moreover, we show several characterisations of the solutions of the extension problems associated to operators with discrete spectrum, namely Laplacians on compact Lie groups, Hermite and special Hermite operators.

Citation: Pradeep Boggarapu, Luz Roncal, Sundaram Thangavelu. On extension problem, trace Hardy and Hardy's inequalities for some fractional Laplacians. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2575-2605. doi: 10.3934/cpaa.2019116
References:
[1]

Adimurthi, P. K. Ratnakumar and V. K. Sohani, A Hardy-Sobolev inequality for the twisted Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), 1-23. doi: 10.1017/S0308210516000081.

[2]

Adimurthi and A. Sekar, Role of the fundamental solution in Hardy-Sobolev-type inequalities, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1111-1130. doi: 10.1017/S030821050000490X.

[3]

V. BanicaM. d. M. González and M. Sáez, Some constructions for the fractional Laplacian on noncompact manifolds, Rev. Mat. Iberoam., 31 (2015), 681-712. doi: 10.4171/RMI/850.

[4]

W. Beckner, Pitt's inequality and the fractional Laplacian: sharp error estimates, Forum Math., 24 (2012), 177-209. doi: 10.1515/form.2011.056.

[5]

K. BogdanB. Dyda and P. Kim, Hardy inequalities and non-explosion results for semigroups, Potential Anal., 44 (2016), 229-247. doi: 10.1007/s11118-015-9507-0.

[6]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monographs in Mathematics, 2007.

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[8]

P. CiattiM. G. Cowling and F. Ricci, Hardy and uncertainty inequalities on stratified Lie groups, Adv. Math., 277 (2015), 365-387. doi: 10.1016/j.aim.2014.12.040.

[9]

R. D. DeBlassie, The first exit time of a two-dimensional symmetric stable process from a wedge, Ann. Probab, 18 (1990), 1034-1070.

[10]

B. DriverL. Gross and L. Saloff-Coste, Holomorphic functions and subelliptic heat kernels over Lie groups, J. Eur. Math. Soc. (JEMS), 11 (2009), 941-978. doi: 10.4171/JEMS/171.

[11]

F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacians in Carnot groups, Math. Z., 279 (2015), 435-458. doi: 10.1007/s00209-014-1376-5.

[12]

S. FilippasL. 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.

[13]

S. FilippasL. Moschini and A. Tertikas, Trace Hardy-Sobolev-Maz'ya inequalities for the half fractional Laplacian, Commun. Pure Appl. Anal., 14 (2015), 373-382. doi: 10.3934/cpaa.2015.14.373.

[14]

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

[15]

G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes, 28. Princeton University Press, N.J.; University of Tokyo Press, Tokyo, 1982.

[16]

R. L. FrankE. H. Lieb and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc., 21 (2008), 925-950. doi: 10.1090/S0894-0347-07-00582-6.

[17]

J. E. GaléP. J. Miana and P. R. Stinga, Extension problem and fractional operators: semigroups and wave equations, J. Evol. Equ., 13 (2013), 343-368. doi: 10.1007/s00028-013-0182-6.

[18]

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.

[19]

B. C. Hall, The Segal-Bargmann "coherent state" transform for compact Lie groups, J. Funct. Anal., 122 (1994), 103-151. doi: 10.1006/jfan.1994.1064.

[20]

B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed., Graduate Texts in Mathematics, 222, Springer, 2015. doi: 10.1007/978-3-319-13467-3.

[21]

I. W. Herbst, Spectral theory of the operator $(p^2+m^2)^{1/2}-Ze^2/r$, Commun. Math. Phys., 53 (1977), 285-294.

[22]

M. Lassalle, L'espace de Hardy d'un domaine de Reinhardt généralisé, J. Funct. Anal., 60 (1985), 309-340. doi: 10.1016/0022-1236(85)90043-6.

[23]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081.

[24]

N. N. Lebedev, Special Functions and Its Applications, Dover, New York, 1972.

[25]

S. A. Molchanov and E. Ostrovskiĭ, Symmetric stable processes as traces of degenerate diffusion processes, Theor. Probab. Appl., 14 (1969), 128-131.

[26]

V. H. Nguyen, Some trace Hardy type inequalities and trace Hardy-Sobolev-Maz'ya type inequalities, J. Funct. Anal., 270 (2016), 4117-4151. doi: 10.1016/j.jfa.2016.03.012.

[27]

L. Roncal and S. Thangavelu, Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group, Adv. Math., 302 (2016), 106-158. doi: 10.1016/j.aim.2016.07.010.

[28]

L. Roncal and S. Thangavelu, An extension problem and trace Hardy inequality for sublaplacians on H-type groups, Int. Math. Res. Not. IMRN., to appear.

[29]

B. Simon, Representations of Finite and Compact Groups. Graduate Studies in Mathematics, 10. American Mathematical Society, Providence, RI, 1996.

[30]

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

[31]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122. doi: 10.1080/03605301003735680.

[32]

G. Szegö, Orthogonal Polynomials, Fourth Edition, Amer. Math. Soc. Colloq. Publ., 23, Amer. Math. Soc., Providence, R. I., 1975.

[33]

S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Mathematical Notes 42. Princeton University Press, Princeton, NJ, 1993.

[34]

S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Progress in Mathematics 159. Birkhäuser, Boston, MA, 1998. doi: 10.1007/978-1-4612-1772-5.

[35]

S. Thangavelu, An introduction to the uncertainty principle. Hardy's theorem on Lie groups. With a foreword by Gerald B. Folland, Progress in Mathematics 217, Birkhäuser, Boston, MA, 2004. doi: 10.1007/978-0-8176-8164-7.

[36]

S. Thangavelu, Gutzmer's formula and Poisson integrals on the Heisenberg group, Pacific J. Math., 231 (2007), 217-237. doi: 10.2140/pjm.2007.231.217.

[37]

S. Thangavelu, On the unreasonable effectiveness of Gutzmer's formula. Harmonic analysis and partial differential equations, 199-217, Contemp. Math., 505, Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/conm/505/09924.

[38]

K. Tzirakis, Improving interpolated Hardy and trace Hardy inequalities on bounded domains, Nonlinear Anal., 127 (2015), 17-34. doi: 10.1016/j.na.2015.06.019.

[39]

H. Urakawa, The heat equation on compact Lie group, Osaka J. Math., 12 (1975), 285-297.

[40]

D. Yafaev, Sharp constants in the Hardy-Rellich inequalities, J. Funct. Anal., 168 (1999), 121-144. doi: 10.1006/jfan.1999.3462.

show all references

References:
[1]

Adimurthi, P. K. Ratnakumar and V. K. Sohani, A Hardy-Sobolev inequality for the twisted Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 147 (2017), 1-23. doi: 10.1017/S0308210516000081.

[2]

Adimurthi and A. Sekar, Role of the fundamental solution in Hardy-Sobolev-type inequalities, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1111-1130. doi: 10.1017/S030821050000490X.

[3]

V. BanicaM. d. M. González and M. Sáez, Some constructions for the fractional Laplacian on noncompact manifolds, Rev. Mat. Iberoam., 31 (2015), 681-712. doi: 10.4171/RMI/850.

[4]

W. Beckner, Pitt's inequality and the fractional Laplacian: sharp error estimates, Forum Math., 24 (2012), 177-209. doi: 10.1515/form.2011.056.

[5]

K. BogdanB. Dyda and P. Kim, Hardy inequalities and non-explosion results for semigroups, Potential Anal., 44 (2016), 229-247. doi: 10.1007/s11118-015-9507-0.

[6]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monographs in Mathematics, 2007.

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[8]

P. CiattiM. G. Cowling and F. Ricci, Hardy and uncertainty inequalities on stratified Lie groups, Adv. Math., 277 (2015), 365-387. doi: 10.1016/j.aim.2014.12.040.

[9]

R. D. DeBlassie, The first exit time of a two-dimensional symmetric stable process from a wedge, Ann. Probab, 18 (1990), 1034-1070.

[10]

B. DriverL. Gross and L. Saloff-Coste, Holomorphic functions and subelliptic heat kernels over Lie groups, J. Eur. Math. Soc. (JEMS), 11 (2009), 941-978. doi: 10.4171/JEMS/171.

[11]

F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacians in Carnot groups, Math. Z., 279 (2015), 435-458. doi: 10.1007/s00209-014-1376-5.

[12]

S. FilippasL. 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.

[13]

S. FilippasL. Moschini and A. Tertikas, Trace Hardy-Sobolev-Maz'ya inequalities for the half fractional Laplacian, Commun. Pure Appl. Anal., 14 (2015), 373-382. doi: 10.3934/cpaa.2015.14.373.

[14]

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

[15]

G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes, 28. Princeton University Press, N.J.; University of Tokyo Press, Tokyo, 1982.

[16]

R. L. FrankE. H. Lieb and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc., 21 (2008), 925-950. doi: 10.1090/S0894-0347-07-00582-6.

[17]

J. E. GaléP. J. Miana and P. R. Stinga, Extension problem and fractional operators: semigroups and wave equations, J. Evol. Equ., 13 (2013), 343-368. doi: 10.1007/s00028-013-0182-6.

[18]

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.

[19]

B. C. Hall, The Segal-Bargmann "coherent state" transform for compact Lie groups, J. Funct. Anal., 122 (1994), 103-151. doi: 10.1006/jfan.1994.1064.

[20]

B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed., Graduate Texts in Mathematics, 222, Springer, 2015. doi: 10.1007/978-3-319-13467-3.

[21]

I. W. Herbst, Spectral theory of the operator $(p^2+m^2)^{1/2}-Ze^2/r$, Commun. Math. Phys., 53 (1977), 285-294.

[22]

M. Lassalle, L'espace de Hardy d'un domaine de Reinhardt généralisé, J. Funct. Anal., 60 (1985), 309-340. doi: 10.1016/0022-1236(85)90043-6.

[23]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081.

[24]

N. N. Lebedev, Special Functions and Its Applications, Dover, New York, 1972.

[25]

S. A. Molchanov and E. Ostrovskiĭ, Symmetric stable processes as traces of degenerate diffusion processes, Theor. Probab. Appl., 14 (1969), 128-131.

[26]

V. H. Nguyen, Some trace Hardy type inequalities and trace Hardy-Sobolev-Maz'ya type inequalities, J. Funct. Anal., 270 (2016), 4117-4151. doi: 10.1016/j.jfa.2016.03.012.

[27]

L. Roncal and S. Thangavelu, Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group, Adv. Math., 302 (2016), 106-158. doi: 10.1016/j.aim.2016.07.010.

[28]

L. Roncal and S. Thangavelu, An extension problem and trace Hardy inequality for sublaplacians on H-type groups, Int. Math. Res. Not. IMRN., to appear.

[29]

B. Simon, Representations of Finite and Compact Groups. Graduate Studies in Mathematics, 10. American Mathematical Society, Providence, RI, 1996.

[30]

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

[31]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122. doi: 10.1080/03605301003735680.

[32]

G. Szegö, Orthogonal Polynomials, Fourth Edition, Amer. Math. Soc. Colloq. Publ., 23, Amer. Math. Soc., Providence, R. I., 1975.

[33]

S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Mathematical Notes 42. Princeton University Press, Princeton, NJ, 1993.

[34]

S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Progress in Mathematics 159. Birkhäuser, Boston, MA, 1998. doi: 10.1007/978-1-4612-1772-5.

[35]

S. Thangavelu, An introduction to the uncertainty principle. Hardy's theorem on Lie groups. With a foreword by Gerald B. Folland, Progress in Mathematics 217, Birkhäuser, Boston, MA, 2004. doi: 10.1007/978-0-8176-8164-7.

[36]

S. Thangavelu, Gutzmer's formula and Poisson integrals on the Heisenberg group, Pacific J. Math., 231 (2007), 217-237. doi: 10.2140/pjm.2007.231.217.

[37]

S. Thangavelu, On the unreasonable effectiveness of Gutzmer's formula. Harmonic analysis and partial differential equations, 199-217, Contemp. Math., 505, Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/conm/505/09924.

[38]

K. Tzirakis, Improving interpolated Hardy and trace Hardy inequalities on bounded domains, Nonlinear Anal., 127 (2015), 17-34. doi: 10.1016/j.na.2015.06.019.

[39]

H. Urakawa, The heat equation on compact Lie group, Osaka J. Math., 12 (1975), 285-297.

[40]

D. Yafaev, Sharp constants in the Hardy-Rellich inequalities, J. Funct. Anal., 168 (1999), 121-144. doi: 10.1006/jfan.1999.3462.

[1]

Stathis Filippas, Luisa Moschini, Achilles Tertikas. Trace Hardy--Sobolev--Maz'ya inequalities for the half fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (2) : 373-382. doi: 10.3934/cpaa.2015.14.373

[2]

Boumediene Abdellaoui, Fethi Mahmoudi. An improved Hardy inequality for a nonlocal operator. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1143-1157. doi: 10.3934/dcds.2016.36.1143

[3]

José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138

[4]

Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951

[5]

Daniele Cassani, Bernhard Ruf, Cristina Tarsi. On the capacity approach to non-attainability of Hardy's inequality in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 245-250. doi: 10.3934/dcdss.2019017

[6]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987

[7]

Mikko Kemppainen, Peter Sjögren, José Luis Torrea. Wave extension problem for the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4905-4929. doi: 10.3934/dcds.2015.35.4905

[8]

Aleksandra Čižmešija, Iva Franjić, Josip Pečarić, Dora Pokaz. On a family of means generated by the Hardy-Littlewood maximal inequality. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 223-231. doi: 10.3934/naco.2012.2.223

[9]

Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935

[10]

Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121

[11]

Pablo Raúl Stinga, Chao Zhang. Harnack's inequality for fractional nonlocal equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3153-3170. doi: 10.3934/dcds.2013.33.3153

[12]

Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171

[13]

Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057

[14]

Lorenzo Brasco, Eleonora Cinti. On fractional Hardy inequalities in convex sets. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4019-4040. doi: 10.3934/dcds.2018175

[15]

Felipe Riquelme. Ruelle's inequality in negative curvature. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2809-2825. doi: 10.3934/dcds.2018119

[16]

S. S. Dragomir, C. E. M. Pearce. Jensen's inequality for quasiconvex functions. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 279-291. doi: 10.3934/naco.2012.2.279

[17]

James Scott, Tadele Mengesha. A fractional Korn-type inequality. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3315-3343. doi: 10.3934/dcds.2019137

[18]

Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934/cpaa.2013.12.1363

[19]

Chunhua Wang, Jing Yang. Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1603-1628. doi: 10.3934/dcds.2016.36.1603

[20]

Jann-Long Chern, Yong-Li Tang, Chuan-Jen Chyan, Yi-Jung Chen. On the uniqueness of singular solutions for a Hardy-Sobolev equation. Conference Publications, 2013, 2013 (special) : 123-128. doi: 10.3934/proc.2013.2013.123

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (29)
  • HTML views (116)
  • Cited by (0)

[Back to Top]