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April  2022, 21(4): 1385-1416. doi: 10.3934/cpaa.2022023

Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

1. 

Università degli Studi di Modena e Reggio Emilia, Dipartimento di Scienze Fisiche, Informatiche e Matematiche, via Campi 213/b, 41125 Modena, Italy

2. 

Università di Napoli Federico II, Dipartimento di Ingegneria Elettrica e delle Tecnologie dell'Informazione, Via Claudio 25, 80125 Napoli, Italy

*Corresponding author

Received  September 2021 Published  April 2022 Early access  January 2022

Fund Project: 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)

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 $
.
Citation: Sergio Polidoro, Annalaura Rebucci, Bianca Stroffolini. Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1385-1416. doi: 10.3934/cpaa.2022023
References:
[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.

[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.

show all references

References:
[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.

[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.

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