An Ontology of Directional Regularity Implying Joint Regularity

Abstract

It is an old idea to consider whether a function on $\mathbb{R}^N$ that is smooth in each variable separately is in fact jointly smooth. It turns out that some uniformity of estimates in each variable is necessary for such a result. More recently, there have been studies of functions that are smooth along integral curves of certain vector fields. Depending on the commutator properties of the vector fields, different types of results may be obtained. Another recent idea is that if one has smoothness along all curves then the uniformity hypothesis may be dropped. In the present paper we explore all these approaches to the problem in a variety of new norms. We present new, simpler proofs of some classical results. We also explore new theorems in the real analytic category.