LAUSR

Abstract 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 ⊂ C n + 1 and M ′ ⊂ C n ′ + 1 with n ′ > n ≥ 1 and M strongly pseudoconvex, we prove that every CR transversal map of class C n ′ − 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 C n ′ − 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 ⊂ C n + 1 and Ω ′ ⊂ C n ′ + 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 C n ′ − 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.