### 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

**On the C ^{∞} regularity of CR mappings of positive codimension.** / Lamel, Bernhard; Mir, Nordine.

Research output: Contribution to journal › Article

^{∞}regularity of CR mappings of positive codimension',

*Advances in Mathematics*, vol. 335, pp. 696-734. https://doi.org/10.1016/j.aim.2018.07.004

^{∞}regularity of CR mappings of positive codimension. Advances in Mathematics. 2018 Sep 7;335:696-734. https://doi.org/10.1016/j.aim.2018.07.004

}

TY - JOUR

T1 - On the C∞ regularity of CR mappings of positive codimension

AU - Lamel, Bernhard

AU - Mir, Nordine

PY - 2018/9/7

Y1 - 2018/9/7

N2 - 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.

AB - 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.

KW - C regularity

KW - CR map

KW - Points of finite type

UR - http://www.scopus.com/inward/record.url?scp=85050095829&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85050095829&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2018.07.004

DO - 10.1016/j.aim.2018.07.004

M3 - Article

VL - 335

SP - 696

EP - 734

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -