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Tytuł artykułu

Ground-state representation for the fractional Laplacian on the half-line

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Języki publikacji
EN
Abstrakty
EN
We give a ground-state representation for the fractional Laplacian with Dirichlet condition on the half-line
Rocznik
Strony
83--108
Opis fizyczny
Bibliogr. 30 poz.
Twórcy
  • Wrocław University of Science and Technology, 27 Wybrzeże Wyspiańskiego St. 50-370 Wrocław, Poland
  • Wrocław University of Science and Technology, 27 Wybrzeże Wyspiańskiego St. 50-370 Wrocław, Poland
Bibliografia
  • [1] B. Abdellaoui, M. Medina, I. Peral, and A. Primo, The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian, J. Differential Equations 260 (2016), 8160-8206.
  • [2] B. Abdellaoui, M. Medina, I. Peral, and A. Primo, Optimal results for the fractional heat equation involving the Hardy potential, Nonlinear Anal. 140 (2016), 166-207.
  • [3] P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984), 121-139.
  • [4] W. Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), 1897-1905.
  • [5] A. BenAmor, The heat equation for the Dirichlet fractional Laplacian with Hardy’s potentials: properties of minimal solutions and blow-up, Rev. Roumaine Math. Pures Appl. 66 (2021), 43-66.
  • [6] K. Bogdan and T. Byczkowski, Potential theory of Schrödinger operator based on fractional Laplacian, Probab. Math. Statist. 20 (2000), 293-335.
  • [7] K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song, and Z. Vondraček, Potential Analysis of Stable Processes and its Extensions, Lecture Notes in Math. 1980, Springer, Berlin, 2009.
  • [8] K. Bogdan, K. Burdzy, and Z. Q. Chen, Censored stable processes, Probab. Theory Related Fields 127 (2003), 89-152.
  • [9] K. Bogdan and B. Dyda, The best constant in a fractional Hardy inequality, Math. Nachr. 284 (2011), 629-638.
  • [10] K. Bogdan, B. Dyda, and P. Kim, Hardy inequalities and non-explosion results for semigroups, Potential Anal. 44 (2016), 229-247.
  • [11] K. Bogdan, T. Grzywny, T. Jakubowski, and D. Pilarczyk, Fractional Laplacian with Hardy potential, Comm. Partial Differential Equations 44 (2019), 20-50.
  • [12] K. Bogdan, T. Grzywny, and M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, Ann. Probab. 38 (2010), 1901-1923.
  • [13] K. Bogdan, T. Grzywny, and M. Ryznar, Density and tails of unimodal convolution semigroups, J. Funct. Anal. 266 (2014), 3543-3571.
  • [14] K. Bogdan and K. Merz, Ground state representation for the fractional Laplacian with Hardy potential in angular momentum channels, arXiv:2305.00881v1 (2023).
  • [15] S. Cho, P. Kim, R. Song, and Z. Vondraˇcek, Factorization and estimates of Dirichlet heat kernels for non-local operators with critical killings, J. Math. Pures Appl. (9) 143, 208-256 (2020).
  • [16] K. L. Chung and Z. X. Zhao, From Brownian Motion to Schrödinger’s Equation, Grundlehren Math. Wiss. 312, Springer, Berlin, 1995.
  • [17] 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.
  • [18] R. L. Frank, K. Merz, and H. Siedentop, Equivalence of Sobolev norms involving generalized Hardy operators, Int. Math. Res. Notices 2021, 2284-2303.
  • [19] R. L. Frank and R. Seiringer, Sharp fractional Hardy inequalities in half-spaces, in: Around the Research of Vladimir Maz’ya. I, Int. Math. Ser. (N.Y.) 11, Springer, New York, 2010, 161-167.
  • [20] M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, de Gruyter Stud. Math. 19, Walter de Gruyter, Berlin, 2011.
  • [21] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., Elsevier/Academic Press, Amsterdam, 7 edition, 2007.
  • [22] G. H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920), 314-317.
  • [23] I. W. Herbst, Spectral theory of the operator (p2 + m2)1/2 − Ze2/r, Comm. Math. Phys. 53 (1977), 285-294.
  • [24] N. Ikeda and S. Watanabe, On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes, J. Math. Kyoto Univ. 2 (1962), 79-95.
  • [25] T. Jakubowski and J. Wang, Heat kernel estimates of fractional Schrödinger operators with negative Hardy potential, Potential Anal. 53 (2020), 997-1024.
  • [26] M. Kijaczko and J. Lenczewska, Sharp Hardy inequalities for Sobolev-Bregman forms, arXiv:2109.01704v2 (2023).
  • [27] A. Kuznetsov, On extrema of stable processes, Ann. Probab. 39 (2011), 1027-1060.
  • [28] M. Kwaśnicki, Fractional Laplace operator and its properties, in: Handbook of Fractional Calculus with Applications, Vol. 1, De Gruyter, Berlin, 2019, 159-193.
  • [29] S. Machihara, T. Ozawa, and H. Wadade, Remarks on the Hardy type inequalities with remainder terms in the framework of equalities, in: Asymptotic Analysis for Nonlinear Dispersive and Wave Equations, Adv. Stud. Pure Math. 81, Math. Soc. Japan, Tokyo, 2019, 247-258.
  • [30] D. Yafaev, Sharp constants in the Hardy-Rellich inequalities, J. Funct. Anal. 168 (1999), 121-144.
Uwagi
Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-e9a35d98-5a73-4fb0-b66b-d5ff896f17cd
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