Representation theorem for locally defined operators in the space of Whitney differentiable functions (Q833083)

The article deals with locally defined operators in the spaces \(C^m(A)\) of Whitney differentiable functions. Recall that elements of the space \(C^m(A)\) (\(A\) is a nonempty set in \({\mathbb R}^n\)) are families of functions \(f^j:A\to{\mathbb R}\), \(|j| \leq m\), with the following property: for all \(j\), \(|j|\leq m\), \(x_0\in A\) and \(\varepsilon > 0\), there exists a \(\delta > 0\) such that, for all \(x, y \in A\), the inequalities \(\|x - x_0\|, \|y - x_0\| < \delta\) imply \[ \Bigg|f^j(x) - \sum_{|i| \leq m - |j|} \frac{f^{i+j}(y)}{i!} (x - y)^i\Bigg| \leq \varepsilon \|x - y\|^{m-|j|}. \] (The norm in this \(C^m(A)\) is defined as usual.) An operator \(K\) between two such spaces is locally defined if, for all \(\varphi, \psi\) and every open interval \(J \subset {\mathbb R}^n\), the equality \(\varphi|_{J \cap A} = \psi|_{J \cap A}\) implies \((K\varphi)|_{J \cap A} = K(\psi)|_{J \cap A}\). The main result is the following statement: If \(A\) is perfect and an operator \(K:C^m(A) \to C^1(A)\) is locally defined, then there exists a unique function \(h:A\times{\mathbb R}^{S(m-1)}\to{\mathbb R}\) such that \[ \begin{multlined} [K(\varphi)]^{(0,\dots,0)}(x) = h(x,\varphi^{(0,\dots,0)}(x),\varphi^{(1,\dots,0)}(x),\dots,\varphi^{(0,\dots,1)}(x),\dots, \\ \varphi^{(m-1,\dots,0)}(x),\dots,\varphi^{(0,\dots,m-1)}(x))\end{multlined} \] for all \(\varphi\in C^m(A)\) and \(x\in A\) (with \(S(m-1) = \sum_{i=0}^{m-1} \binom{n+i-1} {i})\). The authors remark that the function \(h:A\times {\mathbb R}^{S(m-1)} \to {\mathbb R}\) in general is not continuous.