Hardy type inequalities for the fractional relativistic operator

Luz Roncal; Luz Roncal

doi:10.3934/mine.2022018

Mathematics in Engineering

2022, Volume 4, Issue 3: 1-16. doi: 10.3934/mine.2022018

Previous Article Next Article

Research article Special Issues

Hardy type inequalities for the fractional relativistic operator

Luz Roncal ^{1,2,3
,
,}

1.
BCAM - Basque Center for Applied Mathematics 48009 Bilbao, Spain
2.
Ikerbasque, Basque Foundation for Science, 48011 Bilbao, Spain
3.
Universidad del País Vasco/Euskal Herriko Unibertsitatea, 48080 Bilbao, Spain

Received: 17 February 2020 Accepted: 21 April 2020 Published: 30 July 2021

We prove Hardy type inequalities for the fractional relativistic operator by using two different techniques. The first approach goes through trace Hardy inequalities. In order to get the latter, we study the solutions of the associated extension problem. The second develops a non-local version of the ground state representation in the spirit of Frank, Lieb, and Seiringer.
- trace Hardy inequality,
- Hardy inequality,
- extension problem,
- ground state representation,
- fractional operator
Citation: Luz Roncal. Hardy type inequalities for the fractional relativistic operator[J]. Mathematics in Engineering, 2022, 4(3): 1-16. doi: 10.3934/mine.2022018

Related Papers:

Abstract

We prove Hardy type inequalities for the fractional relativistic operator by using two different techniques. The first approach goes through trace Hardy inequalities. In order to get the latter, we study the solutions of the associated extension problem. The second develops a non-local version of the ground state representation in the spirit of Frank, Lieb, and Seiringer.

References

[1]	B. Abdellaoui, M. Medina, I. Peral, A. Primo, Optimal results for the fractional heat equation involving the Hardy potential, Nonlinear Anal., 140 (2016), 166–207. doi: 10.1016/j.na.2016.03.013
[2]	B. Abdellaoui, M. Medina, I. Peral, A. Primo, The effect of the Hardy potential in some Calderón–Zygmund properties for the fractional Laplacian, J. Differ. Equations, 260 (2016), 8160–8206. doi: 10.1016/j.jde.2016.02.016
[3]	P. Baras, J. Goldstein, The heat equation with a singular potential, T. Am. Math. Soc., 294 (1984), 121–139.
[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. Bogdan, B. Dyda, P. Kim, Hardy inequalities and non-explosion results for semigroups, Potential Anal., 44 (2016), 229–247. doi: 10.1007/s11118-015-9507-0
[6]	K. Bogdan, T. Grzywny, T. Jakubowski, D. Pilarczyk, Fractional Laplacian with Hardy potential, Commun. Part. Diff. Eq., 44 (2019), 20–50. doi: 10.1080/03605302.2018.1539102
[7]	P. Boggarapu, L. Roncal, S. Thangavelu, On extension problem, trace Hardy and Hardy's inequalities for some fractional Laplacians, Commun. Pure Appl. Anal., 18, 2575–2605.
[8]	L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Part. Diff. Eq., 32 (2007), 1245–1260. doi: 10.1080/03605300600987306
[9]	M. M. Fall, V. Felli, Sharp essential self-adjointness of relativistic Schrödinger operators with a singular potential, J. Funct. Anal., 267 (2014), 1851–1877. doi: 10.1016/j.jfa.2014.06.010
[10]	M. M. Fall, V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35 (2015), 5827–5867. doi: 10.3934/dcds.2015.35.5827
[11]	R. L. Frank, E. H. Lieb, R. Seiringer, Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators, J. Am. Math. Soc., 21 (2008), 925–950.
[12]	R. L. Frank, R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407–3430. doi: 10.1016/j.jfa.2008.05.015
[13]	J. Fröhlich, B. L. G. Jonsson, E. Lenzmann, Boson stars as solitary waves, Commun. Math. Phys., 274 (2007), 1–30. doi: 10.1007/s00220-007-0272-9
[14]	J. Fröhlich, E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Commun. Pure Appl. Math., 60 (2007), 1691–1705. doi: 10.1002/cpa.20186
[15]	T. Grzywny, M. Ryznar, Two-sided optimal bounds for Green functions of half-spaces for relativistic $\alpha$-stable process, Potential Anal., 28 (2008), 201–239. doi: 10.1007/s11118-007-9071-3
[16]	G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge: Cambridge University Press, 1988.
[17]	I. W. Herbst, Spectral theory of the operator $(p^2+m^2)^{1/2}-Ze^2/r$, Commun. Math. Phys., 53 (1977), 285–294. doi: 10.1007/BF01609852
[18]	L. Hörmander, The analysis of linear partial differential operators I: Distribution Theory and Fourier analysis, 2 Eds., Berlin: Springer-Verlag, 1990.
[19]	N. N. Lebedev, Special functions and its applications, New York: Dover, 1972.
[20]	J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193–248. doi: 10.1007/BF02547354
[21]	E. H. Lieb, The stability of matter: from atoms to stars, B. Am. Math. Soc., 22 (1990), 1–49. doi: 10.1090/S0273-0979-1990-15831-8
[22]	I. Peral, J. L. Vázquez, On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term, Arch. Rational Mech. Anal., 129 (1995), 201–224. doi: 10.1007/BF00383673
[23]	M. Picone, Sui valori eccezionali di un parametro da cui dipende un'equazione differenziale lineare ordinaria del second'ordine, Ann. Scuola Norm. Pisa Cl. Sci., 11 (1910), 144.
[24]	L. Roncal, D. Stan, L. Vega, Carleman type inequalities for fractional relativistic operators, arXiv: 1909.10065.
[25]	L. Roncal, S. Thangavelu, An extension problem and trace Hardy inequality for the sublaplacian on $H$-type groups, Int. Math. Res. Notices, 14 (2020), 4238–4294.
[26]	M. Ryznar, Estimates of Green function for relativistic $\alpha$-stable process, Potential Anal., 17 (2002), 1–23. doi: 10.1023/A:1015231913916
[27]	E. M. Stein, Singular integrals and differentiability properties of functions, New York: Princeton, 1970.
[28]	P. R. Stinga, J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Commun. Part. Diff. Eq., 35 (2010), 2092–2122. doi: 10.1080/03605301003735680
[29]	D. Yafaev, Sharp constants in the Hardy-Rellich inequalities, J. Funct. Anal., 168 (1999), 121–144. doi: 10.1006/jfan.1999.3462

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)