Advanced Search
Article Contents
Article Contents

Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

  • *Corresponding author

    *Corresponding author

This research was partially supported by the grant of Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

Abstract Full Text(HTML) Related Papers Cited by
  • We study the regularity properties of the second order linear operator in $ {{\mathbb {R}}}^{N+1} $:

    $ \begin{equation*} \mathscr{L} u : = \sum\limits_{j,k = 1}^{m} a_{jk}\partial_{x_j x_k}^2 u + \sum\limits_{j,k = 1}^{N} b_{jk}x_k \partial_{x_j} u - \partial_t u, \end{equation*} $

    where $ A = \left( a_{jk} \right)_{j,k = 1, \dots, m}, B = \left( b_{jk} \right)_{j,k = 1, \dots, N} $ are real valued matrices with constant coefficients, with $ A $ symmetric and strictly positive. We prove that, if the operator $ {\mathscr{L}} $ satisfies Hörmander's hypoellipticity condition, and $ f $ is a Dini continuous function, then the second order derivatives of the solution $ u $ to the equation $ {\mathscr{L}} u = f $ are Dini continuous functions as well. We also consider the case of Dini continuous coefficients $ a_{jk} $'s. A key step in our proof is a Taylor formula for classical solutions to $ {\mathscr{L}} u = f $ that we establish under minimal regularity assumptions on $ u $.

    Mathematics Subject Classification: 35K70, 35K65, 35B65.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] F. Anceschi and S. Polidoro, A survey on the classical theory for Kolmogorov equation, Matematiche (Catania), 75 (2020), 221-258.  doi: 10.4418/2020.75.1.11.
    [2] G. ArenaA. O. Caruso and A. Causa, Taylor formula on step two Carnot groups, Rev. Mat. Iberoam., 26 (2010), 239-259.  doi: 10.4171/RMI/600.
    [3] A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, in Springer Monographs in Mathematics, Springer, Berlin, 2007.
    [4] A. Bonfiglioli, Taylor formula for homogeneous groups and applications, Math. Z., 262 (2009), 255-279.  doi: 10.1007/s00209-008-0372-z.
    [5] C. Bucur and A. L. Karakhanyan, Potential theoretic approach to Schauder estimates for the fractional Laplacian, Proc. Amer. Math. Soc., 145 (2017), 637-651.  doi: 10.1090/proc/13227.
    [6] M. Di Francesco and S. Polidoro, Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form, Adv. Differ. Equ., 11 (2006), 1261-1320. 
    [7] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207.  doi: 10.1007/BF02386204.
    [8] G. B. Folland and E. M. Stein, Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group, Commun. Pure Appl. Math., 27 (1974), 429-522.  doi: 10.1002/cpa.3160270403.
    [9] G. B. Folland and E. M. Stein, Hardy spaces on homogeneous groups, in Mathematical Notes, Princeton University Press, 1982.
    [10] La rs Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.
    [11] C. Imbert and L. Silvestre, Regularity for the Boltzmann equation conditional to macroscopic bounds, EMS Press, 7 (2020), 117-172.  doi: 10.4171/emss/37.
    [12] A. Kolmogoroff, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. Math., 35 (1934), 116-117.  doi: 10.2307/1968123.
    [13] E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Partial differential equations, II (Turin, 1993), 52 (1994), 29-63. 
    [14] L. Lorenzi, Schauder estimates for degenerate elliptic and parabolic problems with unbounded coefficients in ${\mathbb R}^N$, Differ. Integral Equ., 18 (2005), 531-566. 
    [15] A. Lunardi, Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in ${\mathbb{R} }^n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 133-164. 
    [16] M. Manfredini, The Dirichlet problem for a class of ultraparabolic equations, Adv. Differ. Equ., 2 (1997), 831-866. 
    [17] S. Menozzi, Parametrix techniques and martingale problems for some degenerate Kolmogorov equations, Electron. Commun. Probab., 16 (2011), 234-250.  doi: 10.1214/ECP.v16-1619.
    [18] A. NagelE. M. Stein and S. Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math., 155 (1985), 103-147.  doi: 10.1007/BF02392539.
    [19] S. PagliaraniA. Pascucci and M. Pignotti, Intrinsic Taylor formula for Kolmogorov-type homogeneous groups, J. Math. Anal. Appl., 435 (2016), 1054-1087.  doi: 10.1016/j.jmaa.2015.10.080.
    [20] Stefano Pagliarani and Michele Pignotti, Intrinsic Taylor formula for non-homogeneous Kolmogorov-type Lie groups, arXiv: 1707.01422v2.
    [21] A. Pascucci, PDE and Martingale Methods in Option Pricing, Milano, Springer, 2011. doi: 10.1007/978-88-470-1781-8. doi: 10.1007/978-88-470-1781-8.
    [22] M. Pignotti, Averaged Stochastic Processes and Kolmogorov Operators, Ph.D thesis, Alma Mater Studiorum Università di Bologna, 2018.
    [23] E. Priola, Global Schauder estimates for a class of degenerate Kolmogorov equations, Stud. Math., 194 (2007), 117-153.  doi: 10.4064/sm194-2-2.
    [24] L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247-320.  doi: 10.1007/BF02392419.
    [25] X. J. Wang, Schauder estimates for elliptic and parabolic equations, Chinese Ann. Math. Ser. B, 27 (2006), 637-642.  doi: 10.1007/s11401-006-0142-3.
    [26] N. WeiY. Jiang and Y. Wu, Partial Schauder estimates for a sub-elliptic equation, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 945-956.  doi: 10.1016/S0252-9602(16)30051-0.
  • 加载中

Article Metrics

HTML views(475) PDF downloads(280) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint