On Boman's Theorem On Partial Regularity Of Mappings

Abstract

Let {\Lambda}\subsetR^{n}\timesR^{m} and k be a positive integer. Let f:R^{n}\rightarrowR^{m} be a locally bounded map such that for each ({\xi},{\eta})\in{\Lambda}, the derivatives D_{{\xi}}^{j}f(x):=|((d^{j})/(dt^{j}))f(x+t{\xi})|_{t=0}, j=1,2,...k, exist and are continuous. In order to conclude that any such map f is necessarily of class C^{k} it is necessary and sufficient that {\Lambda} be not contained in the zero-set of a nonzero homogenous polynomial {\Phi}({\xi},{\eta}) which is linear in {\eta}=({\eta}_{1},{\eta}_{2},...,{\eta}_{m}) and homogeneous of degree k in {\xi}=({\xi}_{1},{\xi}_{2},...,{\xi}_{n}). This generalizes a result of J. Boman for the case k=1. The statement and the proof of a theorem of Boman for the case k=\infty is also extended to include the Carleman classes C{M_{k}} and the Beurling classes C(M_{k}).