On the C∞ regularity of CR mappings of positive codimension

The present paper tackles the C∞ regularity problem for CR maps h:M→M′ between C∞-smooth CR submanifolds M,M′ embedded in complex spaces of possibly different dimensions. For real hypersurfaces M⊂Cn+1 and M′⊂Cn′+1 with n′>n≥1 and M strongly pseudoconvex, we prove that every CR transversal map of class Cn′−n+1 that is nowhere C∞ on some non-empty open subset of M must send this open subset to the set of D'Angelo infinite points of M′. As a corollary, we obtain that every CR transversal map h:M→M′ of class Cn′−n+1 must be C∞-smooth on a dense open subset of M when M′ is of D'Angelo finite type. Another consequence establishes the following boundary regularity result for proper holomorphic maps in positive codimension: given Ω⊂Cn+1 and Ω′⊂Cn′+1 pseudoconvex domains with smooth boundaries ∂Ω and ∂Ω′ both of D'Angelo finite type, n′>n≥1, any proper holomorphic map h:Ω→Ω′ that extends Cn′−n+1-smoothly up to ∂Ω must be C∞-smooth on a dense open subset of ∂Ω. More generally, for CR submanifolds M and M′ of higher codimensions, our main result describes the impact of the existence of a nowhere smooth CR map h:M→M′ on the CR geometry of M′, allowing to extend the previously mentioned results in the hypersurface case to any codimension, as well as deriving a number of regularity results for CR maps with D'Angelo infinite type targets.