# American Institute of Mathematical Sciences

September  2020, 19(9): 4699-4726. doi: 10.3934/cpaa.2020192

## Hardy inequalities for the fractional powers of the Grushin operator

 1 School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710129, China 2 Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Valparaíso, Chile 3 Department of Mathematics, Jianghan University, Wuhan, Hubei, 430056, China

* Corresponding author

Received  August 2019 Revised  March 2020 Published  June 2020

Fund Project: M. Song is supported by the China Postdoctoral Science Foundation (Grant No. 2017M623230), the National Natural Science Foundation of China (Grant No. 11701452), the Natural Science Foundation of Shaanxi Province (Grant No. 2020JQ-112) and the Fundamental Research Funds for the Central Universities (Grant No. 3102018zy041). J. Tan is partially supported by Chile Government grant FONDECYT grant 1160519 and by MINECO grants MTM2014-52402-C3-1-P and MTM2017-84214-C2-1-P (Spain Government Grants)

We establish uncertainty principles and Hardy inequalities for the fractional Grushin operator, which are reduced to those inequalities for the fractional generalized sublaplacian. The key ingredients to obtain them are an explicit integral representation and a ground state representation for the fractional powers of generalized sublaplacian.

Citation: Manli Song, Jinggang Tan. Hardy inequalities for the fractional powers of the Grushin operator. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4699-4726. doi: 10.3934/cpaa.2020192
##### References:
 [1] R. Balhara, Hardy's inequality for the fractional powers of the Grushin operator, Proc. Indian Acad. Sci. (Math. Sci.), 129 (2019), 33. doi: 10.1007/s12044-019-0471-2.  Google Scholar [2] 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 [3] O. Ciaurri, L. Roncal and S. Thangavelu, Hardy-type inequalities for fractional powers of the Dunkl-Hermite operator, Proc. Edinb. Math. Soc., 61 (2018), 513-544.  doi: 10.1017/s0013091517000311.  Google Scholar [4] M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of simple Lie group of real rank one, Invent. Math., 96 (1989), 507-549.  doi: 10.1007/BF01393695.  Google Scholar [5] G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes, Vol. 28, Princeton University Press/University of Tokyo Press, Princeton, NJ/Tokyo, 1982.  Google Scholar [6] R. L. Frank, E. 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 [7] 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 [8] J. Huang, A heat kernel version of Cowling-Price theorem for the Laguerre hypergroup, Proc. Indian Acad. Sci., 120 (2010), 73-81.  doi: 10.1007/s12044-010-0004-5.  Google Scholar [9] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th edition, Elsevier Academic Press, Amsterdam, 2007.  Google Scholar [10] 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 [11] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (PMS-32), Vol. 32, Princeton University Press, 2016.  Google Scholar [12] K. Stempak, An algebra associated with the generalized sublaplacian, Studia Math., 88 (1988), 245-256.  doi: 10.4064/sm-88-3-245-256.  Google Scholar [13] K. Stempak, Mean summability methods for Laguerre series, Trans. Amer. Math. Soc., 322 (1990), 671-690.  doi: 10.2307/2001720.  Google Scholar [14] J. Tan and X. Yu, Liouville type theorems for nonlinear elliptic equations on extended Grushin manifolds, J. Diff. Equa., 269 (2020), 523-541.   Google Scholar [15] S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Math. Notes, Vol. 42, Princeton University Press, Princeton, NJ, 1993.  Google Scholar [16] S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Progress in Mathematics, Vol. 159, Birkhäuser, Boston, MA, 1998. doi: 10.1007/978-1-4612-1772-5.  Google Scholar [17] S. Thangavelu, An Introduction to the Uncertainty Principle. Hardy's Theorem on Lie Groups, Progress in Mathematics, Vol. 217, Birkhäuser, Boston, MA, 2004. doi: 10.1007/978-0-8176-8164-7.  Google Scholar [18] F. G. Tricomi and A. Erdélyi, The asymptotic expansion of a ratio of Gamma functions, Pacific J. Math., 1 (1951), 133-142.   Google Scholar [19] 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] R. Balhara, Hardy's inequality for the fractional powers of the Grushin operator, Proc. Indian Acad. Sci. (Math. Sci.), 129 (2019), 33. doi: 10.1007/s12044-019-0471-2.  Google Scholar [2] 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 [3] O. Ciaurri, L. Roncal and S. Thangavelu, Hardy-type inequalities for fractional powers of the Dunkl-Hermite operator, Proc. Edinb. Math. Soc., 61 (2018), 513-544.  doi: 10.1017/s0013091517000311.  Google Scholar [4] M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of simple Lie group of real rank one, Invent. Math., 96 (1989), 507-549.  doi: 10.1007/BF01393695.  Google Scholar [5] G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes, Vol. 28, Princeton University Press/University of Tokyo Press, Princeton, NJ/Tokyo, 1982.  Google Scholar [6] R. L. Frank, E. 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 [7] 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 [8] J. Huang, A heat kernel version of Cowling-Price theorem for the Laguerre hypergroup, Proc. Indian Acad. Sci., 120 (2010), 73-81.  doi: 10.1007/s12044-010-0004-5.  Google Scholar [9] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th edition, Elsevier Academic Press, Amsterdam, 2007.  Google Scholar [10] 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 [11] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (PMS-32), Vol. 32, Princeton University Press, 2016.  Google Scholar [12] K. Stempak, An algebra associated with the generalized sublaplacian, Studia Math., 88 (1988), 245-256.  doi: 10.4064/sm-88-3-245-256.  Google Scholar [13] K. Stempak, Mean summability methods for Laguerre series, Trans. Amer. Math. Soc., 322 (1990), 671-690.  doi: 10.2307/2001720.  Google Scholar [14] J. Tan and X. Yu, Liouville type theorems for nonlinear elliptic equations on extended Grushin manifolds, J. Diff. Equa., 269 (2020), 523-541.   Google Scholar [15] S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Math. Notes, Vol. 42, Princeton University Press, Princeton, NJ, 1993.  Google Scholar [16] S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Progress in Mathematics, Vol. 159, Birkhäuser, Boston, MA, 1998. doi: 10.1007/978-1-4612-1772-5.  Google Scholar [17] S. Thangavelu, An Introduction to the Uncertainty Principle. Hardy's Theorem on Lie Groups, Progress in Mathematics, Vol. 217, Birkhäuser, Boston, MA, 2004. doi: 10.1007/978-0-8176-8164-7.  Google Scholar [18] F. G. Tricomi and A. Erdélyi, The asymptotic expansion of a ratio of Gamma functions, Pacific J. Math., 1 (1951), 133-142.   Google Scholar [19] D. Yafaev, Sharp constants in the Hardy–Rellich inequalities, J. Funct. Anal., 168 (1999), 121-144.  doi: 10.1006/jfan.1999.3462.  Google Scholar
 [1] Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020104 [2] Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469 [3] Giuseppe Capobianco, Tom Winandy, Simon R. Eugster. The principle of virtual work and Hamilton's principle on Galilean manifolds. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021002 [4] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [5] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445 [6] Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $p$-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293 [7] Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265 [8] Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268 [9] Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255 [10] Zaihui Gan, Fanghua Lin, Jiajun Tong. On the viscous Camassa-Holm equations with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3427-3450. doi: 10.3934/dcds.2020029 [11] Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $p$-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034 [12] Indranil Chowdhury, Gyula Csató, Prosenjit Roy, Firoj Sk. Study of fractional Poincaré inequalities on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020394 [13] Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020367 [14] Biao Zeng. Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021001 [15] Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321 [16] Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054 [17] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [18] Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443 [19] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [20] Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

2019 Impact Factor: 1.105

## Metrics

• PDF downloads (126)
• HTML views (78)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]