Journé's theorem for \(C^{n,\omega}\) regularity (Q945448)

From the author's abstract: 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 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 \(C^{n,\omega ^{\prime}}\) for \(\omega ^{\prime}\left( t\right) =\int_0^t\frac{\omega\left(\tau\right)}{\tau}\,d\tau\). For \(\omega\left( t\right) =t^{\alpha}\), \(0<\alpha <1\), one recovers Journé's result.