On Boman's Theorem On Partial Regularity Of Mappings

TL;DR

This paper generalizes Boman's theorem on partial regularity of functions, establishing conditions under which local boundedness and derivative continuity imply full regularity, extending to higher derivatives and specific function classes.

Contribution

It extends Boman's theorem to higher derivatives and broader function classes, providing necessary and sufficient conditions for partial regularity to imply full smoothness.

Findings

Characterization of when partial derivatives imply full regularity.

Extension of Boman's theorem to derivatives of order k.

Inclusion of Carleman and Beurling classes in the regularity criteria.

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}).