Local operators on \(C^p\) (Q541256)

Let \(E,F,G\) be Banach spaces and \(X\) be an open subset of \(G\). The authors give a characterization of local (that is, support shrinking or preserving) linear operators \(T:\;C^p(X,E)\to C^q(X,F)\), \(1\leq p,q\leq \infty\). The result extends the theorem by \textit{J. C. Wells} and \textit{C. R. DePrima} [Proc. Am. Math. Soc. 40, 453--457 (1973; Zbl 0269.47024)] devoted to the case \(p=q=\infty\), which, in turn, extended the classical Peetre's theorem (where \(E,F,G\) are finite-dimensional). The case where \(p,q\) are arbitrary but \(\dim G<\infty\), \(E=F=\mathbb R\), was considered by \textit{R. Kantrowitz} and \textit{M. M. Neumann} [Glasg. Math. J. 43, No.~2, 295--309 (2001; Zbl 0994.47030)].