### 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.

Original language | English |
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Pages (from-to) | 696-734 |

Number of pages | 39 |

Journal | Advances in Mathematics |

Volume | 335 |

DOIs | |

Publication status | Published - 7 Sep 2018 |

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### Keywords

- C regularity
- CR map
- Points of finite type

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

^{∞}regularity of CR mappings of positive codimension.

*Advances in Mathematics*,

*335*, 696-734. https://doi.org/10.1016/j.aim.2018.07.004