Theory of Probability and Mathematical Statistics
Subordinated Bessel heat kernels
Krzysztof Bogdan and Konstantin Merz
Link
Abstract: We prove new bounds for Bessel heat kernels and Bessel heat kernels subordinated by stable subordinators. In particular, we provide 3G inequalities in the subordinated case
Keywords: Bessel heat kernel, stable subordinator, 3G inequality
Bibliography: Jorge J. Betancor, Alejandro J. Castro, and Pablo Raúl Stinga, The fractional Bessel equation in Hölder spaces, J. Approx. Theory 184 (2014), 55–99. MR 3218793, DOI 10.1016/j.jat.2014.05.003
Jorge J. Betancor, Eleonor Harboure, Adam Nowak, and Beatriz Viviani, Mapping properties of fundamental operators in harmonic analysis related to Bessel operators, Studia Math. 197 (2010), no. 2, 101–140. MR 2600427, DOI 10.4064/sm197-2-1
Krzysztof Bogdan, Tomasz Grzywny, Tomasz Jakubowski, and Dominika Pilarczyk, Fractional Laplacian with Hardy potential, Comm. Partial Differential Equations 44 (2019), no. 1, 20–50. MR 3933622, DOI 10.1080/03605302.2018.1539102
Krzysztof Bogdan and Tomasz Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys. 271 (2007), no. 1, 179–198. MR 2283957, DOI 10.1007/s00220-006-0178-y
K. Bogdan, T. Jakubowski, and K. Merz, Hardy perturbations of subordinated Bessel heat kernels, arXiv e-prints, page arXiv:2409.02853, September 2024.
Krzysztof Bogdan and Konstantin Merz, Ground state representation for the fractional Laplacian with Hardy potential in angular momentum channels, J. Math. Pures Appl. (9) 186 (2024), 176–204 (English, with English and French summaries). MR 4746984, DOI 10.1016/j.matpur.2024.04.003
K. Bogdan and K. Merz, Heat kernel bounds for the fractional Laplacian with Hardy potential in angular momentum channels. In preparation, 2024.
Krzysztof Bogdan, Andrzej Stós, and PawełSztonyk, Harnack inequality for stable processes on d-sets, Studia Math. 158 (2003), no. 2, 163–198. MR 2013738, DOI 10.4064/sm158-2-5
Krzysztof Bogdan and Karol Szczypkowski, Gaussian estimates for Schrödinger perturbations, Studia Math. 221 (2014), no. 2, 151–173. MR 3200161, DOI 10.4064/sm221-2-4
Andrei N. Borodin and Paavo Salminen, Handbook of Brownian motion—facts and formulae, 2nd ed., Probability and its Applications, Birkhäuser Verlag, Basel, 2002. MR 1912205, DOI 10.1007/978-3-0348-8163-0
F. Bouzeffour and M. Garayev, On the fractional Bessel operator, Integral Transforms Spec. Funct. 33 (2022), no. 3, 230–246. MR 4381553, DOI 10.1080/10652469.2021.1925268
Jacek Dziubański and Marcin Preisner, Hardy spaces for semigroups with Gaussian bounds, Ann. Mat. Pura Appl. (4) 197 (2018), no. 3, 965–987. MR 3802701, DOI 10.1007/s10231-017-0711-y
Rupert L. Frank and Konstantin Merz, On Sobolev norms involving Hardy operators in a half-space, J. Funct. Anal. 285 (2023), no. 10, Paper No. 110104, 54. MR 4634216, DOI 10.1016/j.jfa.2023.110104
Vanesa Galli, Sandra Molina, and Alejandro Quintero, Liouville theorems for the multidimensional fractional Bessel operators, Commun. Korean Math. Soc. 37 (2022), no. 4, 1099–1129. MR 4505564, DOI 10.4134/CKMS.c210361
T. Grzywny and B. Trojan, Subordinated Markov processes: sharp estimates for heat kernels and Green functions, arXiv e-prints, page arXiv:2110.01201, October 2021.
Wolfhard Hansen, Global comparison of perturbed Green functions, Math. Ann. 334 (2006), no. 3, 643–678. MR 2207878, DOI 10.1007/s00208-005-0719-2
Tomasz Jakubowski and Jian Wang, Heat kernel estimates of fractional Schrödinger operators with negative Hardy potential, Potential Anal. 53 (2020), no. 3, 997–1024. MR 4140086, DOI 10.1007/s11118-019-09795-7
Jacek Małecki, Grzegorz Serafin, and Tomasz Zorawik, Fourier-Bessel heat kernel estimates, J. Math. Anal. Appl. 439 (2016), no. 1, 91–102. MR 3474351, DOI 10.1016/j.jmaa.2016.02.051
F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds., NIST Digital Library of Mathematical Functions.
K. A. Penson and K. Górska, Exact and explicit probability densities for one-sided Lévy stable distributions, Phys. Rev. Lett. 105 (2010), no. 21, 210604, 4. MR 2740992, DOI 10.1103/PhysRevLett.105.210604
Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
René L. Schilling, Renming Song, and Zoran Vondraček, Bernstein functions, 2nd ed., De Gruyter Studies in Mathematics, vol. 37, Walter de Gruyter & Co., Berlin, 2012. Theory and applications. MR 2978140, DOI 10.1515/9783110269338
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, NJ, 1971. MR 304972
