Peetre-Slov\'ak's theorem revisited

Abstract

In 1960, J. Peetre proved the finiteness of the order of linear local operators. Later on, J. Slov\'{a}k vastly generalized this theorem, proving the finiteness of the order of a broad class of (non-linear) local operators. In this paper, we use the language of sheaves and ringed spaces to prove a simpler version of Slov\'{a}k's result. The statement we prove, adapting Slov\'{a}k's original ideas, deals with local operators defined between the sheaves of smooth sections of fibre bundles, and thus covers many of the applications of Slov\'{a}k's theorem.