본문 바로가기
KERD
논문KCI 등재

Sobolev estimates for the local extension of boundary holomorphic forms on real hypersurfaces in Cn

조상현
대한수학회 (The Korean Mathematical Society)|2013.05|대한수학회지 (Journal of the KMS)|제50권 제3호|pp. 479-491

2

조회

0

다운로드

0

인용

0

참고문헌

키워드

tangential Cauchy-Riemann equationboundary holomorphic forms
주제분류수학

초록

Let M be a smooth real hypersurface in complex space ofdimension n, n ≥ 3, and assume that the Levi-form at z0 on M has atleast (q+1)-positive eigenvalues, 1 ≤ q ≤ n−2. We estimate solutions ofthe local ∂-closed extension problem near z0 for (p, q)-forms in Sobolevspaces. Using this result, we estimate the local solution of tangentialCauchy-Riemann equation near z0 in Sobolev spaces.

Abstract

Let M be a smooth real hypersurface in complex space ofdimension n, n ≥ 3, and assume that the Levi-form at z0 on M has atleast (q+1)-positive eigenvalues, 1 ≤ q ≤ n−2. We estimate solutions ofthe local ∂-closed extension problem near z0 for (p, q)-forms in Sobolevspaces. Using this result, we estimate the local solution of tangentialCauchy-Riemann equation near z0 in Sobolev spaces.

참고문헌 (23)

  1. A. Andreotti, "E. E. Levi convexity and the Hans Lewy problem. I, II", Ann. Scuola Norm. Sup. Pisa (3), 26, pp.325~363 (1972)
  2. A. Andreotti, "E. E. Levi convexity and the Hans Lewy problem. I, II", Ann. Scuola Norm. Sup. Pisa (3), 26, pp.747~806 (1972)
  3. S. Bell, "Differentiability of the Bergman kernel and pseudo-local estimates", Math. Z, 192, pp.467~472 (1986)
  4. H. P. Boas, "Sobolev estimates for the Lewy operator on weakly pseudo- convex boundaries", Math. Ann, 274(2), pp.221~231 (1986)
  5. A. Boggess, "A kernel approach to the local solvability of the tangential Cauchy Riemann equations", Trans. Amer. Math. Soc, 289(2), pp.643~658 (1985)
  6. D. Catlin, "Sufficient conditions for the extension of CR structures", J. Geom. Anal, 4(4) (1994)
  7. D. Catlin, "Extension of CR structures on three dimensional compact pseu- doconvex CR manifolds", Math. Ann, 334(2), pp.253~280 (2006)
  8. S.-C. Chen, "Partial Differential equations in several complex variables, AMS", IP Studies in Advanced Mathematics, 2001, pp.19, ~[DOI]
  9. S. Cho, "Local extension of boundary holomorphic forms on real hypersurfaces in Cn", J. Math. Anal. Appl, 325(1), pp.279~293 (2007)
  10. 조상현, "Explicit Sobolev estimates for the Cauchy-Riemann equation on parameters", 대한수학회보, 45(2), pp.321~338 (2008), 대한수학회
  11. S. Cho, "Sobolev estimates for the local extension of ∂b-closed (0, 1)-forms on real hyper- surfaces in Cn", Proc. of Amer. Math. Soc, 139(11), pp.4053~4062 (2011)
  12. P. C. Greiner, "Necessary and sufficient conditions for solvability of the Lewy equation", Proc. Nat. Acad. Sci. U.S.A, 72(9), pp.3287~3289 (1975)
  13. R. S. Hamilton, "Deformation of complex structures on manifolds with boundary I, II", J. Diff. Geom, 12(1), pp.1~15 (1977)
  14. R. S. Hamilton, "Deformation of complex structures on manifolds with boundary I, II", J. Diff. Geom, 12(3), pp.409~473 (1979)
  15. G. M. Henkin, "Andreotti-Grauert theory by Integral Formulas" (1988), Birkh¨auser
  16. J. J. Kohn, "On the extension of holomorphic functions from the boundary of a complex manifold", Ann. of Math. (2), 81, pp.451~472 (1965)
  17. M. Kuranishi, "Strongly pseudoconvex CR structures over small balls. I. An a priori estimate", Ann. of Math. (2), 115(3), pp.451~500 (1982)
  18. C. Laurent-Thi´ebaut, "On the Hartogs-Bochner extension phenomenon for differential forms", Math. Ann, 284(1), pp.103~119 (1989)
  19. H. Lewy, "On the local character of the solution of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables", Ann. of Math. (2), 64, pp.514~522 (1956)
  20. H. Lewy, "An example of a smooth linear partial differential equation without solution", Ann. of Math. (2), 66, pp.155~158 (1957)
  21. M.-C. Shaw, "L2 estimates and existence theorems for the tangential Caucy-Riemann complex", Invent. Math, 82(1), pp.133~150 (1985)
  22. M.-C. Shaw, "L2 existence theorems for the ∂b-Neumann problem on strongly pseudoconvex CR manifolds", J. Geom. Anal. (2), 1, pp.139~163 (1992)
  23. S. Webster, "On the proof of Kuranishi’s embedding theorem", Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 6(3), pp.183~207 (1989)

관련 논문AI추천

Explicit Sobolev estimates for the Cauchy-Riemann equation on parameters조상현, 최재서대한수학회 · 대한수학회보2008 · 제45권 제2호A bound on H\"older regularity for $\overline \partial$-equation on pseudoconvex domains in $\mathbb C^n$ with some comparable eigenvalues of the Levi-form조상현대한수학회 · 대한수학회보2021 · 제58권 제3호$L^p$-estimates for the {\large$\bar{\partial}$}-equation with exact support on $q$-convex intersectionsShaban Khidr대한수학회 · 대한수학회지2018 · 제55권 제1호The $\overline\partial$-problem with support conditions and pseudoconvexity of general order in K\"ahler manifoldsSayed Saber대한수학회 · 대한수학회지2016 · 제53권 제6호Solution to $\overline{\partial}$-problem with support conditions in weakly $q$-convex domainsSayed Saber대한수학회 · 대한수학회논문집2018 · 제33권 제2호GLOBAL SOLUTIONS FOR THE a-PROBLEM ON NON PSEUDOCONVEX DOMAINS IN STEIN MANIFOLDSSayed Saber대한수학회 · 대한수학회지2017 · 제54권 제6호

이 간행물의 다른 논문 (11건)

Time scales integral inequalities for superquadratic functionsJosip Pecaric, Rabia Bibi, Josipa Baric, Martin Bohnerpp. 465-477Surfaces of general type with pg=1 and q=0박희상, 신동수, 박종일pp. 493-507Explicit expression of the Krawtchouk polynomial via a discrete Green's function이윤진, 김길천pp. 509-527Avoiding permutations and the Narayana numbers박승경, 박영자pp. 529-541On the minimal graded free resolution of powers of lexsegment idealsAnda Olteanupp. 543-555
원문 보기