Hardy inequalities for the fractional powers of the Grushin operator

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

Hardy inequalities for the fractional powers of the Grushin operator

Manli Song ^1, and Jinggang Tan ^2,3,,

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.

Keywords: Fractional Grushin operator, fractional generalized sublaplacian, Hardy's inequality, uncertainty principle.

Mathematics Subject Classification: Primary: 35H20, 26D10.

Citation: Manli Song, Jinggang Tan. Hardy inequalities for the fractional powers of the Grushin operator. Communications on Pure and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7^th edition, Elsevier Academic Press, Amsterdam, 2007.
[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.
[11]	E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (PMS-32), Vol. 32, Princeton University Press, 2016.
[12]	K. Stempak, An algebra associated with the generalized sublaplacian, Studia Math., 88 (1988), 245-256. doi: 10.4064/sm-88-3-245-256.
[13]	K. Stempak, Mean summability methods for Laguerre series, Trans. Amer. Math. Soc., 322 (1990), 671-690. doi: 10.2307/2001720.
[14]	J. Tan and X. Yu, Liouville type theorems for nonlinear elliptic equations on extended Grushin manifolds, J. Diff. Equa., 269 (2020), 523-541.
[15]	S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Math. Notes, Vol. 42, Princeton University Press, Princeton, NJ, 1993.
[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.
[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.
[18]	F. G. Tricomi and A. Erdélyi, The asymptotic expansion of a ratio of Gamma functions, Pacific J. Math., 1 (1951), 133-142.
[19]	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]	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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7^th edition, Elsevier Academic Press, Amsterdam, 2007.
[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.
[11]	E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (PMS-32), Vol. 32, Princeton University Press, 2016.
[12]	K. Stempak, An algebra associated with the generalized sublaplacian, Studia Math., 88 (1988), 245-256. doi: 10.4064/sm-88-3-245-256.
[13]	K. Stempak, Mean summability methods for Laguerre series, Trans. Amer. Math. Soc., 322 (1990), 671-690. doi: 10.2307/2001720.
[14]	J. Tan and X. Yu, Liouville type theorems for nonlinear elliptic equations on extended Grushin manifolds, J. Diff. Equa., 269 (2020), 523-541.
[15]	S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Math. Notes, Vol. 42, Princeton University Press, Princeton, NJ, 1993.
[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.
[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.
[18]	F. G. Tricomi and A. Erdélyi, The asymptotic expansion of a ratio of Gamma functions, Pacific J. Math., 1 (1951), 133-142.
[19]	D. Yafaev, Sharp constants in the Hardy–Rellich inequalities, J. Funct. Anal., 168 (1999), 121-144. doi: 10.1006/jfan.1999.3462.

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