Hardy type inequalities for the fractional relativistic operator. (English) Zbl 1511.81057
Summary: 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.
MSC:
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 81R20 | Covariant wave equations in quantum theory, relativistic quantum mechanics |
| 47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |
| 30H10 | Hardy spaces |
| 41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |
| 81P16 | Quantum state spaces, operational and probabilistic concepts |
| 35R11 | Fractional partial differential equations |
| 35P15 | Estimates of eigenvalues in context of PDEs |
Keywords:
trace Hardy inequality; Hardy inequality; extension problem; ground state representation; fractional operatorReferences:
| [1] | B, Optimal results for the fractional heat equation involving the Hardy potential, Nonlinear Anal., 140, 166-207 (2016) · Zbl 1383.35238 · doi:10.1016/j.na.2016.03.013 |
| [2] | B, The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian, J. Differ. Equations, 260, 8160-8206 (2016) · Zbl 1386.35422 · doi:10.1016/j.jde.2016.02.016 |
| [3] | P, The heat equation with a singular potential, T. Am. Math. Soc., 294, 121-139 (1984) · Zbl 0556.35063 |
| [4] | W, Pitt’s inequality and the fractional Laplacian: sharp error estimates, Forum Math., 24, 177-209 (2012) · Zbl 1244.42005 · doi:10.1515/form.2011.056 |
| [5] | K, Hardy inequalities and non-explosion results for semigroups, Potential Anal., 44, 229-247 (2016) · Zbl 1335.31007 · doi:10.1007/s11118-015-9507-0 |
| [6] | K, Fractional Laplacian with Hardy potential, Commun. Part. Diff. Eq., 44, 20-50 (2019) · Zbl 07051869 · 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. · Zbl 1481.35017 |
| [8] | L, An extension problem related to the fractional Laplacian, Commun. Part. Diff. Eq., 32, 1245-1260 (2007) · Zbl 1143.26002 · doi:10.1080/03605300600987306 |
| [9] | M, Sharp essential self-adjointness of relativistic Schrödinger operators with a singular potential, J. Funct. Anal., 267, 1851-1877 (2014) · Zbl 1295.35377 · doi:10.1016/j.jfa.2014.06.010 |
| [10] | M, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35, 5827-5867 (2015) · Zbl 1336.35356 · doi:10.3934/dcds.2015.35.5827 |
| [11] | R, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, J. Am. Math. Soc., 21, 925-950 (2008) · Zbl 1202.35146 |
| [12] | R, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255, 3407-3430 (2008) · Zbl 1189.26031 · doi:10.1016/j.jfa.2008.05.015 |
| [13] | J, Boson stars as solitary waves, Commun. Math. Phys., 274, 1-30 (2007) · Zbl 1126.35064 · doi:10.1007/s00220-007-0272-9 |
| [14] | J, Blowup for nonlinear wave equations describing boson stars, Commun. Pure Appl. Math., 60, 1691-1705 (2007) · Zbl 1135.35011 · doi:10.1002/cpa.20186 |
| [15] | T, Two-sided optimal bounds for Green functions of half-spaces for relativistic \(\alpha \)-stable process, Potential Anal., 28, 201-239 (2008) · Zbl 1146.60059 · doi:10.1007/s11118-007-9071-3 |
| [16] | G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge: Cambridge University Press, 1988. · Zbl 0634.26008 |
| [17] | I, Spectral theory of the operator \((p^2+m^2)^{1/2}-Ze^2/r\), Commun. Math. Phys., 53, 285-294 (1977) · Zbl 0375.35047 · 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. · Zbl 0712.35001 |
| [19] | N. N. Lebedev, Special functions and its applications, New York: Dover, 1972. · Zbl 0271.33001 |
| [20] | J, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, 193-248 (1934) · JFM 60.0726.05 · doi:10.1007/BF02547354 |
| [21] | E, The stability of matter: from atoms to stars, B. Am. Math. Soc., 22, 1-49 (1990) · Zbl 0698.35135 · doi:10.1090/S0273-0979-1990-15831-8 |
| [22] | I, On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term, Arch. Rational Mech. Anal., 129, 201-224 (1995) · Zbl 0821.35080 · doi:10.1007/BF00383673 |
| [23] | M, Sui valori eccezionali di un parametro da cui dipende un’equazione differenziale lineare ordinaria del second’ordine, Ann. Scuola Norm. Pisa Cl. Sci., 11, 144 (1910) · JFM 41.0351.01 |
| [24] | L. Roncal, D. Stan, L. Vega, Carleman type inequalities for fractional relativistic operators, arXiv: 1909.10065. |
| [25] | L, An extension problem and trace Hardy inequality for the sublaplacian on \(H\)-type groups, Int. Math. Res. Notices, 14, 4238-4294 (2020) · Zbl 1484.35015 |
| [26] | M, Estimates of Green function for relativistic \(\alpha \)-stable process, Potential Anal., 17, 1-23 (2002) · Zbl 1004.60047 · doi:10.1023/A:1015231913916 |
| [27] | E. M. Stein, Singular integrals and differentiability properties of functions, New York: Princeton, 1970. · Zbl 0207.13501 |
| [28] | P, Extension problem and Harnack’s inequality for some fractional operators, Commun. Part. Diff. Eq., 35, 2092-2122 (2010) · Zbl 1209.26013 · doi:10.1080/03605301003735680 |
| [29] | D, Sharp constants in the Hardy-Rellich inequalities, J. Funct. Anal., 168, 121-144 (1999) · Zbl 0981.26016 · doi:10.1006/jfan.1999.3462 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
