Uniqueness of derivatives of functions defined on closed sets (Q1072061)

Let \(\Lambda_{\alpha}({\mathbb{R}}^ n)\), \(\alpha >0\), \(k<\alpha \leq k+1\), k integer, denote the Lipschitz space consisting of k times continuously differentiable functions f satisfying \[ | D^ jf(x+h)-2D^ jf(x)+D^ jf(x-h)| \leq c| h|^{\alpha -k},\quad x,h\in R^ n,\quad | j| =k\quad and\quad | f(x)| \leq c,\quad x\in R^ n. \] The problem studied in the paper can be formulated in terms of Whitney derivatives, but it can also be stated as follows: Let \(1\leq k<\alpha \leq k+1\). For which closed sets F is it true that if \(f\in \Lambda_{\alpha}({\mathbb{R}}^ n)\) and \(f(x)=0\), \(x\in F\), then \(D^ jf(x)=0\), \(| j| \leq k?\) It is shown (Corollary 1) that sets F with this property may be characterized as follows. There is no point \(x_ 0\in F\), such that there exists an open neighbourhood U of \(x_ 0\) and an (n-1)-dimensional surface M of class \(\Lambda_{\alpha -k+1}\) with \(F\cap U\subset M\cap U\). The proof is based on a theorem on multiplication of functions in the Lipschitz spaces \(\Lambda_{\alpha}(F)\) (Theorem 1) and a version of the implicit function theorem. The corresponding problem for the spaces \(C^ k({\mathbb{R}}^ n)\) has been studied by \textit{G. Glaeser} [J. Analyse Math. 6, 1-124 (1958; Zbl 0091.281)].