We consider a class of perturbations of the degenerate Ornstein-Uhlenbeck operator in . Using a revised version of Bernstein’s method we provide several uniform estimates for the semigroup associated with the realization of the operator in the space of all the bounded and continuous functions in
@article{ASNSP_2005_5_4_2_255_0, author = {Lorenzi, Luca}, title = {Estimates of the derivatives for a class of parabolic degenerate operators with unbounded coefficients in $\mathbb {R}^N$}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {255--293}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 4}, number = {2}, year = {2005}, mrnumber = {2163557}, zbl = {1107.35071}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2005_5_4_2_255_0/} }
TY - JOUR AU - Lorenzi, Luca TI - Estimates of the derivatives for a class of parabolic degenerate operators with unbounded coefficients in $\mathbb {R}^N$ JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2005 SP - 255 EP - 293 VL - 4 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2005_5_4_2_255_0/ LA - en ID - ASNSP_2005_5_4_2_255_0 ER -
%0 Journal Article %A Lorenzi, Luca %T Estimates of the derivatives for a class of parabolic degenerate operators with unbounded coefficients in $\mathbb {R}^N$ %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2005 %P 255-293 %V 4 %N 2 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2005_5_4_2_255_0/ %G en %F ASNSP_2005_5_4_2_255_0
Lorenzi, Luca. Estimates of the derivatives for a class of parabolic degenerate operators with unbounded coefficients in $\mathbb {R}^N$. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 4 (2005) no. 2, pp. 255-293. http://archive.numdam.org/item/ASNSP_2005_5_4_2_255_0/
[1] Sur la généralisation du probléme de Dirichlet, I, Math. Ann. 62 (1906), 253-271. | JFM | MR
,[2] Analytic methods for Markov semigroups, Preprint 401, Dipartimento di Matematica, Università di Parma, 2005. | MR
and ,[3] Estimates of the derivatives for parabolic operators with unbounded coefficients, Trans. Amer. Math. Soc. (to appear). | MR | Zbl
and ,[4] Some results for second order elliptic operators having unbounded coefficients, Differential Integral Equations 11 (1998), 561-588. | MR | Zbl
,[5] Regularity results for some degenerate parabolic equations, Riv. Mat. Univ. Parma (6) 2* (1999), 245-257. | MR | Zbl
,[6] Gradient estimates for Dirichlet parabolic problems in unbounded domains, J. Differential Equations 205 (2004), 329-353. | MR | Zbl
, and ,[7] “Partial Differential Equations of Parabolic Type”, Prentice Hall, Englewood Cliffs, N.J., 1964. | MR | Zbl
,[8] “Stochastic Stability of Differential Equations”, Nauka 1969 (in Russian), English translation: Sijthoff and Noordhoff 1980. | MR
,[9] “Introduction to the Theory of Diffusion Processes”, American Mathematical Society 142, (1992). | MR | Zbl
,[10] “Linear and Quasilinear Equations of Parabolic Type”, Nauka, English transl.: American Mathematical Society, Providence, 1968. | Zbl
, and ,[11] “Second Order Parabolic Differential Equations”, World Scientific Publishing Co. Pte. Ltd, Singapore, New Jersey, London Hong Kong, 1996. | MR | Zbl
,[12] Schauder estimates for a class of degenerate elliptic and parabolic problems with unbounded coefficients, Differential Integral Equations 18 (2005), 531-566. | MR
,[13] “Analytic Semigroups and Optimal Regularity in Parabolic Problems”, Birkhäuser, Basel, 1995. | MR | Zbl
,[14] Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), 133-164. | Numdam | MR | Zbl
,[15] Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in , Studia Math. 128 (1998), 171-198. | MR | Zbl
,[16] The Dirichlet problem for a class of ultraparabolic equations, Adv. Differential Equations 2 (1997), 831-866. | MR | Zbl
,[17] A priori estimates for quasilinear degenerate parabolic equations, Proc. Amer. Math. Soc. 131 (2002), 1115-1120. | MR | Zbl
and ,[18] Feller semigroups on , Semigroup Forum 65 (2002), 159-205. | MR | Zbl
, and ,[19] Hölder regularity for a Kolmogorov equation, Trans. Amer. Math. Soc. 355 (2002), 901-924. | MR | Zbl
,[20] On a class of ultraparabolic operators of Kolmogorov-Fokker-Plank type, Matematiche (Catania) 49 (1994), 53-105 (1995). | MR | Zbl
,[21] The Cauchy problem for a class of Markov-type semigroups, Comm. Appl. Anal. 5 (2001), 49-75. | MR | Zbl
,