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

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 $0<\beta$.