[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.2013.50.3.479

SOBOLEV ESTIMATES FOR THE LOCAL EXTENSION OF BOUNDARY HOLOMORPHIC FORMS ON REAL HYPERSURFACES IN ℂⁿ

Cho, Sanghyun (Department of Mathematics Sogang University)

Publication Information

Journal of the Korean Mathematical Society / v.50, no.3, 2013 , pp. 479-491 More about this Journal

Abstract

Let M be a smooth real hypersurface in complex space of dimension $n$ , $n{\geq}3$ , and assume that the Levi-form at $z_0$ on M has at least $(q+1)$ -positive eigenvalues, $1{\leq}q{\leq}n-2$ . We estimate solutions of the local $\bar{\partial}$ -closed extension problem near $z_0$ for $(p,q)$ -forms in Sobolev spaces. Using this result, we estimate the local solution of tangential Cauchy-Riemann equation near $z_0$ in Sobolev spaces.

Keywords

tangential Cauchy-Riemann equation; boundary holomorphic forms;

Citations & Related Records

Times Cited By KSCI : 1 (Citation Analysis)

Reference
Cited By KSCI

1	A. Andreotti and C. D. Hill, E. E. Levi convexity and the Hans Lewy problem. I, II, Ann. Scuola Norm. Sup. Pisa (3) 26 (1972), 325-363
2	A. Andreotti and C. D. Hill, E. E. Levi convexity and the Hans Lewy problem. I, II, Ann. Scuola Norm. Sup. Pisa (3) 26 (1972), 747-806.
3	S. Bell, Differentiability of the Bergman kernel and pseudo-local estimates, Math. Z. 192 (1986), 467-472. DOI
4	H. P. Boas and M.-C. Shaw, Sobolev estimates for the Lewy operator on weakly pseudo-convex boundaries, Math. Ann. 274 (1986), no. 2, 221-231. DOI
5	A. Boggess and M.-C. Shaw, A kernel approach to the local solvability of the tangential Cauchy Riemann equations, Trans. Amer. Math. Soc. 289 (1985), no. 2, 643-658. DOI ScienceOn
6	D. Catlin, Sufficient conditions for the extension of CR structures, J. Geom. Anal. 4 (1994), no. 4, 467-538. DOI ScienceOn
7	D. Catlin and S. Cho, Extension of CR structures on three dimensional compact pseu-doconvex CR manifolds, Math. Ann. 334 (2006), no. 2, 253-280. DOI
8	S.-C. Chen and M.-C. Shaw, Partial Differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, 19. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001.
9	S. Cho and J. Choi, Local extension of boundary holomorphic forms on real hypersurfaces in $C^n$ , J. Math. Anal. Appl. 325 (2007), no. 1, 279-293. DOI ScienceOn
10	S. Cho, Explicit Sobolev estimates for the Cauchy-Riemann equation on parameters, Bull. Korean Math. Soc. 45 (2008), no. 2, 321-338. 과학기술학회마을 DOI ScienceOn
11	S. Cho, Sobolev estimates for the local extension of $\overline{\partial}_b$ -closed (0, 1)-forms on real hyper- surfaces in $C^n$ , Proc. of Amer. Math. Soc. 139 (2011), no. 11, 4053-4062.
12	P. C. Greiner, J. J. Kohn, and E. M. Stein, Necessary and sufficient conditions for solvability of the Lewy equation, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), no. 9, 3287-3289. DOI ScienceOn
13	R. S. Hamilton, Deformation of complex structures on manifolds with boundary I, II, J. Diff. Geom. 12 (1977), no. 1, 1-15
14	R. S. Hamilton, Deformation of complex structures on manifolds with boundary I, II, J. Diff. Geom. 14 (1979), no. 3, 409-473. DOI
15	G. M. Henkin and J. Leiterer, Andreotti-Grauert theory by Integral Formulas, Birkhauser, Boston, Basel, 1988.
16	J. J. Kohn and H. Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. of Math. (2) 81 (1965), 451-472. DOI
17	M. Kuranishi, Strongly pseudoconvex CR structures over small balls. I. An a priori estimate, Ann. of Math. (2) 115 (1982), no. 3, 451-500. DOI
18	H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math. (2) 66 (1957), 155-158. DOI
19	C. Laurent-Thiebaut and J. Leiterer, On the Hartogs-Bochner extension phenomenon for differential forms, Math. Ann. 284 (1989), no. 1, 103-119. DOI
20	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 (1956), 514-522. DOI
21	M.-C. Shaw, $L^2$ estimates and existence theorems for the tangential Caucy-Riemann complex, Invent. Math. 82 (1985), no. 1, 133-150. DOI
22	M.-C. Shaw, $L^2$ existence theorems for the $\overline{\\partial}_b$ -Neumann problem on strongly pseudoconvex CR manifolds, J. Geom. Anal. (2) 1 (1992), 139-163.
23	S. Webster, On the proof of Kuranishi's embedding theorem, Ann. Inst. H. Poincare Anal. Non Lineaire 6 (1989), no. 3, 183-207. DOI

KSCI

SOBOLEV ESTIMATES FOR THE LOCAL EXTENSION OF BOUNDARY HOLOMORPHIC FORMS ON REAL HYPERSURFACES IN ℂn

SOBOLEV ESTIMATES FOR THE LOCAL EXTENSION OF BOUNDARY HOLOMORPHIC FORMS ON REAL HYPERSURFACES IN ℂⁿ