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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[29]

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

[30]

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

[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.  Google Scholar

[32]

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

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[40]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[29]

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

[30]

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

[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.  Google Scholar

[32]

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

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[40]

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

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