Differentiability of functions: approximate, global, and differentiability along curves over non-archimedean fields

Abstract

This paper is devoted to the study of approximate and global smoothness and smoothness along curves of functions f(x 1,...,x m ) of variables x 1,...,x m in infinite fields with nontrivial non-Archimedean valuations and relations between them. Theorems on classes of smoothness C n or \(C_{\rm{b}}^n\) of functions with partial difference quotients continuous or bounded uniformly continuous on bounded domains up to order n are investigated. We prove that from f ○ u ∈ C n(K, K l) or f ○ u ∈ \(C_{\rm{b}}^n\)(K, K l) for each C ∞ or \(C_{\rm {b}}^\infty\) curve u: K → K m it follows that f ∈ C n(K m, K l) or f ∈ \(C_{\rm{b}}^n\)(K m, K l), where m ≥ 2. Then the classes of smoothness C n,r and \(C_{\rm{b}}^{n,r}\) and more general in the sense of Lipschitz for partial difference quotients are considered and theorems for them are proved. Moreover, the approximate differentiability of functions relative to measures is defined and investigated. Its relations with the Lipschitzian property and almost everywhere differentiability are studied. Non-Archimedean analogs of classical theorems of Kirzsbraun, Rademacher, Stepanoff, and Whitney are formulated and proved, and substantial differences between two cases are found. Finally, theorems about relations between approximate differentiability by all variables and along curves are proved.