Smoothness of functions global and along curves over ultra-metric fields

2006/08/29 by S. V. Ludkovsky, Ludkovsky, S. V.
Mathematics · #46S10 #58C20 #58C25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:46S10 #msc:58C20 #msc:58C25

paper · pdf · doi:10.48550/arxiv.math/0608725

39 pages

arxiv created 2007/03/27 · arxiv updated 2009/12/01

Abstract

The article is devoted to the investigation of smoothness of functions f(x₁,...,x_m) of variables x₁,...,x_m in infinite fields with non-trivial multiplicative ultra-norms, where m≥ 2. Theorems about classes of smoothness Cⁿ or Cⁿ_b of functions with continuous or bounded uniformly continuous on bounded domains partial difference quotients up to the order n are investigated. It is proved, that from f∘ u∈ Cⁿ(\bf K,\bf K^l) or f∘ u∈ Cⁿ_b(\bf K,\bf K^l) for each C^∞ or C^∞ _b curve u: \bf K→ \bf K^m it follows, that f∈ Cⁿ(\bf K^m,\bf K^l) or f∈ Cⁿ_b(\bf K^m,\bf K^l) respectively. Moreover, classes of smoothness C^n,r and C^n,r_b and more general in the sense of Lipschitz for partial difference quotients are considered and theorems for them are proved.

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