Local Boundedness of Catlin q-Type


YAZICI Ö.

MEDITERRANEAN JOURNAL OF MATHEMATICS, vol.19, no.1, 2022 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 19 Issue: 1
  • Publication Date: 2022
  • Doi Number: 10.1007/s00009-021-01945-9
  • Title of Journal : MEDITERRANEAN JOURNAL OF MATHEMATICS
  • Keywords: Catlin type, order of contact, germs of holomorphic functions, subelliptic estimates, NEUMANN PROBLEM, ORDERS

Abstract

In D'Angelo (Ann Math 115:615-637, 1982) introduced the notion of finite type for points p of a real hypersurface M of C-n by defining the order of contact Delta(q)(M, p) of complex-analytic q-dimensional varieties with M at p. Later, Catlin (Ann Math 126(1):131-191, 1987) defined q-type, D-q(M, p) for points of hypersurfaces by considering generic (n - q + 1)-dimensional complex aline subspaces of C-n. We define a generalization of the Catlin's q-type for an arbitrary subset M of C-n in a similar way that D'Angelo's 1-type, (M, p), is generalized in Lamel and Mir (Adv Math 335:696-734, 2018). Using recent results connecting the D'Angelo and Catlin q-types in Brinzanescu and Nicoara (Relating Catlin and D'Angelo q-types. arXiv:1707.08294, 2021) and building on D'Angelo's work on the openness of the set of points of finite Delta(q)-type, we prove the openness of the set of points of finite Catlin q-type for an arbitrary subset M subset of C-n.