Global Sobolev regularity and pointwise upper bounds for the gradient of transition densities associated with second order elliptic operators in $ \mathbb{R}^d $ was obtained in the case of bounded diffusion coefficients in [13, Section 5]. In this paper we generalize these results to the case of unbounded diffusions. Our technique is based on an approximation procedure and a De Giorgi type regularity result. We like to point out, that such an approximation procedure cannot be applied to the result in [13], since the constants in the estimates obtained in [13] depend on the infinity norm of the diffusion coefficients.
Citation: |
[1] |
A. Aibeche, K. Laidoune and A. Rhandi, Time dependent Lyapunov functions for some Kolmogorov semigroups perturbed by unbounded potentials, Arch. Math. (Basel), 94 (2010), 565-577.
doi: 10.1007/s00013-010-0124-2.![]() ![]() ![]() |
[2] |
W. Arendt, G. Metafune and D. Pallara, Gaussian estimates for elliptic operators with unbounded drift, J. Math. Anal. Appl., 338 (2008), 505-517.
doi: 10.1016/j.jmaa.2007.05.006.![]() ![]() ![]() |
[3] |
V. I. Bogachev, M. Rëkner and S. V. Shaposhnikov, Global regularity and estimates for solutions of parabolic equations, Teor. Veroyatn. Primen., 50 (2005), 652-674.
![]() ![]() |
[4] |
V. I. Bogachev, N. V. Krylov, M. Röckner and S. V. Shaposhnikov, Fokker-Planck-Kolmogorov Equations, vol. 207 of Mathematical surveys and monographs, American Mathematical Society, Providence, Rhode Island, 2015.
doi: 10.1090/surv/207.![]() ![]() ![]() |
[5] |
V. I. Bogachev and S. V. Shaposhnikov, Representations of solutions to Fokker-Planck-Kolmogorov equations with coefficients of low regularity, J. Evol. Equ., 20 (2020), 355-374.
doi: 10.1007/s00028-019-00532-6.![]() ![]() ![]() |
[6] |
L. Caso, M. Kunze, M. Porfido and A. Rhandi, General kernel estimates of Schrödinger type operators with unbounded diffusion terms, To appear in Proc. A Royal Soc. Edinburgh.
doi: 10.48550/arXiv.2204.12146.![]() ![]() |
[7] |
M. Kunze, L. Lorenzi and A. Rhandi, Kernel estimates for nonautonomous Kolmogorov equations with potential term, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, vol. 10 of Springer INdAM Ser., Springer, Cham, 2014, 229-251.
doi: 10.1007/978-3-319-11406-4_12.![]() ![]() ![]() |
[8] |
M. Kunze, L. Lorenzi and A. Rhandi, Kernel estimates for nonautonomous Kolmogorov equations, Adv. Math., 287 (2016), 600-639.
doi: 10.1016/j.aim.2015.09.029.![]() ![]() ![]() |
[9] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967 (in Russian); Transl. Math. Monographs 23, AMS, Providence, RI, 1967 (in English).
![]() ![]() |
[10] |
K. Laidoune, G. Metafune, D. Pallara and A. Rhandi, Global properties of transition kernels associated with second-order elliptic operators, in Parabolic Problems, vol. 80 of Progr. Nonlinear Differential Equations Appl., Birkhäuser/Springer Basel AG, Basel, 2011,415-432.
doi: 10.1007/978-3-0348-0075-4_21.![]() ![]() ![]() |
[11] |
L. Lorenzi and M. Bertoldi, Analytical Methods for Markov Semigroups, vol. 283 of Pure and Applied Mathematics (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2007.
![]() ![]() |
[12] |
L. Lorenzi and A. Rhandi, Semigroups of Bounded Operators and Second-Order Elliptic and
Parabolic Partial Differential Equations, Monographs and Research Notes in Mathematics,
CRC Press, Boca Raton, FL, 2021.
doi: 10.1201/9780429262593.![]() ![]() ![]() |
[13] |
G. Metafune, D. Pallara and A. Rhandi, Global properties of transition probabilities of singular diffusions, Teor. Veroyatn. Primen., 54 (2009), 116-148.
doi: 10.4213/tvp2549.![]() ![]() ![]() |
[14] |
G. Metafune, D. Pallara and M. Wacker, Compactness properties of Feller semigroups, Studia Math., 153 (2002), 179-206.
doi: 10.4064/sm153-2-5.![]() ![]() ![]() |
[15] |
M. Porfido, Estimates for the Transition Kernel for Elliptic Operators with Unbounded Coefficients (Doctoral Thesis), University of Salerno, Italy, 2023.
![]() |
[16] |
S. V. Shaposhnikov, Fokker-Planck-Kolmogorov equations with a potential and a nonuniformly elliptic diffusion matrix, Tr. Mosk. Mat. Obs., 74 (2013), 15-29.
doi: 10.1090/S0077-1554-2014-00211-9.![]() ![]() ![]() |
[17] |
C. Spina, Kernel estimates for a class of Kolmogorov semigroups, Arch. Math. (Basel), 91 (2008), 265-279.
doi: 10.1007/s00013-008-2676-y.![]() ![]() ![]() |