The complex Green operator on CR-submanifolds of ℂⁿ of hypersurface type: Compactness

Straube, Emil

doi:10.1090/S0002-9947-2012-05510-3

The complex Green operator on CR-submanifolds of $\mathbb {C}^{n}$ of hypersurface type: Compactness
HTML articles powered by AMS MathViewer

by Emil J. Straube PDF

Trans. Amer. Math. Soc. 364 (2012), 4107-4125 Request permission

Abstract:

We establish compactness estimates for $\overline {\partial }_{b}$ on a compact pseudoconvex CR-submanifold of $\mathbb {C}^{n}$ of hypersurface type that satisfies property(P). When the submanifold is orientable, these estimates were proved by A. Raich in 2010 using microlocal methods. Our proof deduces the estimates from (a slight extension, when $q>1$, of) those known on hypersurfaces via the fact that locally, CR-submanifolds of hypersurface type are CR-equivalent to a hypersurface. The relationship between two potential theoretic conditions is also clarified.

References

Heungju Ahn, Global boundary regularity for the $\overline \partial$-equation on $q$-pseudoconvex domains, Math. Nachr. 280 (2007), no. 4, 343–350. MR 2294804, DOI 10.1002/mana.200410486
É. Amar, Cohomologie complexe et applications, J. London Math. Soc. (2) 29 (1984), no. 1, 127–140 (French, with English summary). MR 734998, DOI 10.1112/jlms/s2-29.1.127
M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series, vol. 47, Princeton University Press, Princeton, NJ, 1999. MR 1668103, DOI 10.1515/9781400883967
Steve Bell, Differentiability of the Bergman kernel and pseudolocal estimates, Math. Z. 192 (1986), no. 3, 467–472. MR 845219, DOI 10.1007/BF01164021
Harold P. Boas and Mei-Chi Shaw, Sobolev estimates for the Lewy operator on weakly pseudoconvex boundaries, Math. Ann. 274 (1986), no. 2, 221–231. MR 838466, DOI 10.1007/BF01457071
Harold P. Boas and Emil J. Straube, Sobolev estimates for the complex Green operator on a class of weakly pseudoconvex boundaries, Comm. Partial Differential Equations 16 (1991), no. 10, 1573–1582. MR 1133741, DOI 10.1080/03605309108820813
Albert Boggess, CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991. MR 1211412
Boutet de Monvel, L. Intégration des équations de Cauchy-Riemann induites formelles, Séminaire Goulaouic-Lions-Schwartz 1974–1975; Équations Aux Derivées Partielles Linéaires et Non Linéaires, Exp. No. 9, Centre Math., École Polytech., Paris, 1975.
Daniel M. Burns Jr., Global behavior of some tangential Cauchy-Riemann equations, Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977) Lecture Notes in Pure and Appl. Math., vol. 48, Dekker, New York, 1979, pp. 51–56. MR 535588
David W. Catlin, Global regularity of the $\bar \partial$-Neumann problem, Complex analysis of several variables (Madison, Wis., 1982) Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, RI, 1984, pp. 39–49. MR 740870, DOI 10.1090/pspum/041/740870
Çelik, Mehmet, Contributions to the compactness theory of the $\overline {\partial }$-Neumann operator, Diss. Texas A&M University, 2008.
Mehmet Çelik and Emil J. Straube, Observations regarding compactness in the $\overline {\partial }$-Neumann problem, Complex Var. Elliptic Equ. 54 (2009), no. 3-4, 173–186. MR 2513533, DOI 10.1080/17476930902759478
So-Chin Chen and Mei-Chi Shaw, Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. MR 1800297, DOI 10.1090/amsip/019
E. B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. MR 1349825, DOI 10.1017/CBO9780511623721
G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Annals of Mathematics Studies, No. 75, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0461588
Fu, Siqi, private communication
Siqi Fu and Emil J. Straube, Compactness of the $\overline \partial$-Neumann problem on convex domains, J. Funct. Anal. 159 (1998), no. 2, 629–641. MR 1659575, DOI 10.1006/jfan.1998.3317
Siqi Fu and Emil J. Straube, Compactness in the $\overline \partial$-Neumann problem, Complex analysis and geometry (Columbus, OH, 1999) Ohio State Univ. Math. Res. Inst. Publ., vol. 9, de Gruyter, Berlin, 2001, pp. 141–160. MR 1912737
Roger Gay and Ahmed Sebbar, Division et extension dans l’algèbre $A^\infty (\Omega )$ d’un ouvert pseudo-convexe à bord lisse de $\textbf {C}^n$, Math. Z. 189 (1985), no. 3, 421–447 (French). MR 783566, DOI 10.1007/BF01164163
Harrington, Phillip S. and Raich, Andrew, Regularity results for $\overline {\partial }_{b}$ on CR-manifolds of hypersurface type, Commun. Partial Diff. Equations, to appear.
F. Reese Harvey and H. Blaine Lawson Jr., On boundaries of complex analytic varieties. I, Ann. of Math. (2) 102 (1975), no. 2, 223–290. MR 425173, DOI 10.2307/1971032
Lars Hörmander, $L^{2}$ estimates and existence theorems for the $\bar \partial$ operator, Acta Math. 113 (1965), 89–152. MR 179443, DOI 10.1007/BF02391775
Khanh, Tran V., A general method of weights in the $\overline {\partial }$-Neumann problem, preprint 2010, arXiv:1001.5093.
Kenneth D. Koenig, A parametrix for the $\overline \partial$-Neumann problem on pseudoconvex domains of finite type, J. Funct. Anal. 216 (2004), no. 2, 243–302. MR 2095685, DOI 10.1016/j.jfa.2004.06.004
J. J. Kohn, Boundary regularity of $\bar \partial$, Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979) Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, N.J., 1981, pp. 243–260. MR 627761
J. J. Kohn, The range of the tangential Cauchy-Riemann operator, Duke Math. J. 53 (1986), no. 2, 525–545. MR 850548, DOI 10.1215/S0012-7094-86-05330-5
J. J. Kohn, Embedding of CR manifolds, Partial differential equations and related subjects (Trento, 1990) Pitman Res. Notes Math. Ser., vol. 269, Longman Sci. Tech., Harlow, 1992, pp. 151–162. MR 1190939
J. J. Kohn, Superlogarithmic estimates on pseudoconvex domains and CR manifolds, Ann. of Math. (2) 156 (2002), no. 1, 213–248. MR 1935846, DOI 10.2307/3597189
J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443–492. MR 181815, DOI 10.1002/cpa.3160180305
Jeffery D. McNeal, A sufficient condition for compactness of the $\overline \partial$-Neumann operator, J. Funct. Anal. 195 (2002), no. 1, 190–205. MR 1934357, DOI 10.1006/jfan.2002.3958
Samangi Munasinghe and Emil J. Straube, Complex tangential flows and compactness of the $\overline {\partial }$-Neumann operator, Pacific J. Math. 232 (2007), no. 2, 343–354. MR 2366358, DOI 10.2140/pjm.2007.232.343
Andreea C. Nicoara, Global regularity for $\overline \partial _b$ on weakly pseudoconvex CR manifolds, Adv. Math. 199 (2006), no. 2, 356–447. MR 2189215, DOI 10.1016/j.aim.2004.12.006
Andrew Raich, Compactness of the complex Green operator on CR-manifolds of hypersurface type, Math. Ann. 348 (2010), no. 1, 81–117. MR 2657435, DOI 10.1007/s00208-009-0470-1
Andrew S. Raich and Emil J. Straube, Compactness of the complex Green operator, Math. Res. Lett. 15 (2008), no. 4, 761–778. MR 2424911, DOI 10.4310/MRL.2008.v15.n4.a13
Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
Şahutoğlu, Sönmez, Strong Stein neighborhood bases, preprint, 2009.
Mei-Chi Shaw, Global solvability and regularity for $\bar \partial$ on an annulus between two weakly pseudoconvex domains, Trans. Amer. Math. Soc. 291 (1985), no. 1, 255–267. MR 797058, DOI 10.1090/S0002-9947-1985-0797058-3
Mei-Chi Shaw, $L^2$-estimates and existence theorems for the tangential Cauchy-Riemann complex, Invent. Math. 82 (1985), no. 1, 133–150. MR 808113, DOI 10.1007/BF01394783
—, The closed range property for $\overline {\partial }$ on domains with pseudoconcave boundary, In Complex Analysis: Several Complex Variables and Connections with PDEs and Geometry (Fribourg 2008). Trends in Mathematics, Birkhäuser Verlag, 2010.
Nessim Sibony, Une classe de domaines pseudoconvexes, Duke Math. J. 55 (1987), no. 2, 299–319 (French). MR 894582, DOI 10.1215/S0012-7094-87-05516-5
Emil J. Straube, Plurisubharmonic functions and subellipticity of the $\overline \partial$-Neumann problem on non-smooth domains, Math. Res. Lett. 4 (1997), no. 4, 459–467. MR 1470417, DOI 10.4310/MRL.1997.v4.n4.a2
Emil J. Straube, Geometric conditions which imply compactness of the $\overline \partial$-Neumann operator, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 3, 699–710 (English, with English and French summaries). MR 2097419
Emil J. Straube, Lectures on the $\scr L^2$-Sobolev theory of the $\overline {\partial }$-Neumann problem, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2010. MR 2603659, DOI 10.4171/076
W. J. Sweeney, A condition for subellipticity in Spencer’s Neumann problem, J. Differential Equations 21 (1976), no. 2, 316–362. MR 413194, DOI 10.1016/0022-0396(76)90125-X
J.-M. Trépreau, Sur le prolongement holomorphe des fonctions C-R défines sur une hypersurface réelle de classe $C^2$ dans $\textbf {C}^n$, Invent. Math. 83 (1986), no. 3, 583–592 (French). MR 827369, DOI 10.1007/BF01394424
Giuseppe Zampieri, Complex analysis and CR geometry, University Lecture Series, vol. 43, American Mathematical Society, Providence, RI, 2008. MR 2400390, DOI 10.1090/ulect/043