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 $0<\beta$.