Smoothness via directional smoothness and Marchaud's theorem in Banach spaces

Abstract

Classical Marchaud's theorem (1927) asserts that if f is a bounded function on [a,b], k∈N, and the (k+1)th modulus of smoothness ωk+1(f;t) is so small that η(t)=∫0tωk+1(f;s)sk+1ds<+∞ for t>0, then f∈Ck((a,b)) and f(k) is uniformly continuous with modulus cη for some c>0 (i.e. in our terminology f is Ck,cη-smooth). Using a known version of the converse of Taylor theorem we easily deduce Marchaud's theorem for functions on certain open connected subsets of Banach spaces from the classical one-dimensional version. In the case of a bounded subset of Rn our result is more general than that of H. Johnen and K. Scherer (1973), which was proved by quite a different method. We also prove that if a locally bounded mapping between Banach spaces is Ck,ω-smooth on every line, then it is Ck,cω-smooth for some c>0.