An ontology of directional regularity implying joint regularity (Q1047469)

This is a survey on results which state that a function on a subset of \(\mathbb R^n\) has a certain smoothness (Hölder, Lipschitz, higher-order smoothness, belongs to a Sobolev space, or is (real or complex) analytic) if it has this smoothness in each of the variables separately and satisfies some extra conditions. One of these extra conditions is a uniform bound in the corresponding norms, and moreover, it then suffices to assume smoothness along a few paths (even less than \(n\) paths if the paths can be chosen appropriately). Another variant of such results which requires no knowledge of a uniform bound is to assume smoothness along each path from an appropriate class. For complex analytic functions even stronger results hold true. Besides being a survey, the paper also contains shorter new proofs for some of the results and new results in the class of real analytic functions.