Journé's theorem for C^{n,\omega} regularity

Previous Article
Distortion estimates for planar diffeomorphisms
DCDS Home
This Issue
Next Article
Algebro-geometric methods for hard ball systems

January & February 2008, 22(1&2): 413-425. doi: 10.3934/dcds.2008.22.413

Journé's theorem for $C^{n,\omega}$ regularity

1.	Department of Mathematics, West Chester University, West Chester, PA 19383

Received May 2007 Revised January 2008 Published June 2008

Abstract
Related Papers

Let $U$ be an open set in $\mathbb R^2$ and $f:U\to \mathbb R$ a function. $f$ is said to be $C^{n,\alpha}$ if it is $C^n$ and has the $n$-th derivative $\alpha$-Hölder, $0<\alpha< 1$. We generalize a result due to Journé [2] about the $C^{n,\alpha}$ regularity of a real valued continuous function on $U$ that is $C^{n,\alpha}$ along two transverse continuous foliations with $C^{n,\alpha}$ leaves. For $\omega$ a Dini modulus of continuity, $f$ is said to be $C^{n,\omega}$ if it is $C^n$ and has the $n$-th derivative bounded in the seminorm defined by $\omega$. We assume that $f$ is $C^{n,\omega}$ along two transverse continuous foliations with $C^{n,\omega}$ leaves, and show that, under an additional summability condition for the modulus, $f$ is $C^{n,\omega'}$ for $\omega'(t)=\int^t_0\frac{\omega(\tau)}{\tau}d\tau.$ For $\omega(t)=t^{\alpha}, 0<\alpha< 1,$ one recovers Journé's result. Examples of moduli that satisfy our assumptions are given by $\omega(t)=t^{\alpha}( \ln\frac{1}{t} )^{\beta}$, for $0<\alpha<1$ and $0<\beta$.

Keywords: $C^{n, \omega}$ regularity, foliations., Dini modulus of continuity.

Mathematics Subject Classification: Primary: 37D20, 58A05; Secondary: 41A0.

Citation: V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413

[1]	Boris Hasselblatt. Critical regularity of invariant foliations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 931-937. doi: 10.3934/dcds.2002.8.931
[2]	Xinsheng Wang, Lin Wang, Yujun Zhu. Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2125-2140. doi: 10.3934/dcds.2018087
[3]	Wendong Wang, Liqun Zhang. The $C^{\alpha}$ regularity of weak solutions of ultraparabolic equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1261-1275. doi: 10.3934/dcds.2011.29.1261
[4]	Nathan Albin, Nethali Fernando, Pietro Poggi-Corradini. Modulus metrics on networks. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 1-17. doi: 10.3934/dcdsb.2018161
[5]	Linlin Fu, Jiahao Xu. A new proof of continuity of Lyapunov exponents for a class of $ C^2 $ quasiperiodic Schrödinger cocycles without LDT. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2915-2931. doi: 10.3934/dcds.2019121
[6]	Ludovic Rifford. Refinement of the Benoist theorem on the size of Dini subdifferentials. Communications on Pure & Applied Analysis, 2008, 7 (1) : 119-124. doi: 10.3934/cpaa.2008.7.119
[7]	Fengping Yao, Shulin Zhou. Interior $C^{1,\alpha}$ regularity of weak solutions for a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1635-1649. doi: 10.3934/dcdsb.2016015
[8]	Paolo Maremonti. On the Stokes problem in exterior domains: The maximum modulus theorem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2135-2171. doi: 10.3934/dcds.2014.34.2135
[9]	Ethan Akin. On chain continuity. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 111-120. doi: 10.3934/dcds.1996.2.111
[10]	Gabriel Ponce, Ali Tahzibi, Régis Varão. Minimal yet measurable foliations. Journal of Modern Dynamics, 2014, 8 (1) : 93-107. doi: 10.3934/jmd.2014.8.93
[11]	Radu Saghin. Note on homology of expanding foliations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 349-360. doi: 10.3934/dcdss.2009.2.349
[12]	Charles Pugh, Michael Shub, Amie Wilkinson. Hölder foliations, revisited. Journal of Modern Dynamics, 2012, 6 (1) : 79-120. doi: 10.3934/jmd.2012.6.79
[13]	Percy Fernández-Sánchez, Jorge Mozo-Fernández, Hernán Neciosup. Dicritical nilpotent holomorphic foliations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3223-3237. doi: 10.3934/dcds.2018140
[14]	Olha Ivanyshyn. Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Problems & Imaging, 2007, 1 (4) : 609-622. doi: 10.3934/ipi.2007.1.609
[15]	Peng Mei, Zhan Zhou, Genghong Lin. Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2085-2095. doi: 10.3934/dcdss.2019134
[16]	Toshikazu Ito, Bruno Scárdua. Holomorphic foliations transverse to manifolds with corners. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 537-544. doi: 10.3934/dcds.2009.25.537
[17]	Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639
[18]	Fernando Alcalde Cuesta, Ana Rechtman. Minimal Følner foliations are amenable. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 685-707. doi: 10.3934/dcds.2011.31.685
[19]	Yong Fang, Patrick Foulon, Boris Hasselblatt. Zygmund strong foliations in higher dimension. Journal of Modern Dynamics, 2010, 4 (3) : 549-569. doi: 10.3934/jmd.2010.4.549
[20]	Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences