Nonlinear conditions for differentiability of functions (Q1095467)

The paper answers a problem of \textit{H. Joris} [Arch. Math. 39, 269-277 (1982; Zbl 0504.58007)], related to the problem of Taylor (1976) whether \(f^ 2,f^ 3\in C^{\infty}\) implies \(f\in C^{\infty}\) (affirmatively solved by Joris in the quoted paper), and indicates some other questions concerning that field. For instance, the following extensions of the inverse map theorem hold: Let M be a \(C^{\infty}\) real manifold, A a function algebra, either (a) \(\phi_ j: R\to C\) is \(C^{\infty}\) or (b) \(\phi_ j: C\to C\) is complex analytic \((j=1,2,...,k)\) and let \(\phi_ j(k)+\sum^{r_ j(c)}_{\ell =1}a_{j\ell}(c)(t-c)^{n_{j\ell}(c)}\) be a formal Taylor series expansion for \(\phi_ j\) \((0\neq a_{j\ell}(c)\in C\), \(r_ j(c)\in N\cup \{+\infty \}\), \(0<n_{j1}(c)<n_{j2}(c)<...).\) Theorem. Suppose (a) resp. (b) holds, \(f: M\to R\) (resp. \(f: M\to A)\) is continuous and g.c.d. \(\{n_{11}(c)\), \(n_{21}(c),...,n_{k1}(c)\}=1\) for each \(c\in im(f)\) (resp. \(c\in C)\). If \(\phi_ j\circ f\in C^{\infty}\) (in the strong sense, resp.) for \(j=1,2,...,k\), then \(f\in C^{\infty}\) (in the strong sense, resp.). Theorem. Suppose each \(\phi_ j\) is a holomorphic polynomial and g.c.d. \(\{m,n_{11}(0),...,n_{1j_ 1}(0)\), \(n_{21}(0),...,n_{2j_ 2}(0),...,n_{k1}(0),...,n_{kj_ k}(0)\}=1\) for some \(m,j_ 1,j_ 2,...,j_ k\in N\). If \(f: M\to A\) is continuous, \(\phi_ j\circ f\) is \(C^{\infty}\) in the strong sense for \(j=1,2,...,k\) and \(Q\circ f\) is \(C^{\infty}\) in the strong sense where \(Q(z)=z^ m\), then f is \(C^{\infty}\) in the strong sense. In particular, if \(f: R\to R\) is continuous, \(f^ 2+f^ 3\in C^{\infty}\) and \(f^ m\in C\) for some positive integer m, then \(f\in C^{\infty}\).