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Compactness of the complex Green operator on non-pseudoconvex CR manifolds
1. | Universidade Federal de São Carlos, Departamento de Matemática, Rodovia Washington Luis, Km 235 - Caixa Postal 676, São Carlos, Brazil |
2. | Department of Mathematical Sciences, SCEN 327, 1 University of Arkansas, Fayetteville, AR 72701, USA |
In this paper, we investigate the compactness theory of the complex Green operator on smooth, embedded, orientable CR manifolds of hypersurface type that satisfy the weak $ Y(q) $ condition. The sufficient condition that we define is an adaption of the CR-$ P_q $ property for weak $ Y(q) $ manifolds and does not require that the CR manifold is the boundary of a domain.
We also provide several non-pseudoconvex examples (and a level $ q $) for which the complex Green operator is compact.
References:
[1] |
R. Basener,
Nonlinear Cauchy-Riemann equations and $q$-pseudoconvexity, Duke Math. J., 43 (1976), 203-213.
|
[2] |
S. Biard and E. Straube, $L^2$-Sobolev theory for the complex Green operator, Internat. J. Math., 28 (2017), 1740006, 31.
doi: 10.1142/S0129167X17400067. |
[3] |
A. Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, Florida, 1991. |
[4] |
D. Catlin, Global regularity of the $\bar\partial$-Neumann problem, in Complex analysis of several variables (Madison, Wis., 1982), Proc. Sympos. Pure Math., 41, Amer. Math. Soc., Providence, RI, 1984, 39-49.
doi: 10.1090/pspum/041/740870. |
[5] |
S. C. Chen and M. C. Shaw, Partial Differential Equations in Several Complex Variables, vol. 19 of Studies in Advanced Mathematics, American Mathematical Society, 2001.
doi: 10.11650/twjm/1500405913. |
[6] |
J. Coacalle and A. Raich,
Closed range estimates for $\bar\partial_b$ on CR manifolds of hypersurface type, J. Geom. Anal., 31 (2021), 366-394.
doi: 10.1007/s12220-019-00268-2. |
[7] |
R. Diaz,
Necessary conditions for subellipticity of ${\Box}_b$ on pseudoconvex domains, Commun. Partial Differ. Equ., 11 (1986), 1-61.
doi: 10.1080/03605308608820417. |
[8] |
R. Diaz,
Necessary conditions for local subellipticity of $\square_b$ on CR manifolds, J. Differ. Geom., 29 (1989), 389-419.
|
[9] |
G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann Complex, vol. 75 of Ann. of Math. Stud., Princeton University Press, Princeton, New Jersey, 1972. |
[10] |
P. Harrington and A. Raich,
Regularity results for $\bar\partial_b$ on CR-manifolds of hypersurface type, Commun. Partial Differ. Equ., 36 (2011), 134-161.
doi: 10.1080/03605302.2010.498855. |
[11] |
P. Harrington and A. Raich,
Closed range for $\bar\partial$ and $\bar\partial_b$ on bounded hypersurfaces in Stein manifolds, Ann. Inst. Fourier (Grenoble), 65 (2015), 1711-1754.
|
[12] |
P. S. Harrington, M. Peloso and A. Raich,
Regularity equivalence of the Szegö projection and the complex Green operator, Proc. Amer. Math. Soc., 143 (2015), 353-367.
doi: 10.1090/S0002-9939-2014-12393-8. |
[13] |
P. S. Harrington and A. Raich,
Closed range of $ \bar\partial$ in $L^2$-Sobolev spaces on unbounded domains in $ \mathbb C^n$, J. Math. Anal. Appl., 459 (2018), 1040-1461.
doi: 10.1016/j.jmaa.2017.11.017. |
[14] |
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
doi: 10.1017/CBO9780511810817.![]() ![]() ![]() |
[15] |
T. Khanh, S. Pinton and G. Zampieri,
Compactness estimates for $\square_{b}$ on a CR manifold, Proc. Amer. Math. Soc., 140 (2012), 3229-3236.
|
[16] |
K. Koenig,
A parametrix for the $\overline\partial$-Neumann problem on pseudoconvex domains of finite type, J. Funct. Anal., 216 (2004), 243-302.
doi: 10.1016/j.jfa.2004.06.004. |
[17] |
S. Munasinghe and E. Straube,
Geometric sufficient conditions for compactness of the complex Green operator, J. Geom. Anal., 22 (2012), 1007-1026.
doi: 10.1007/s12220-011-9226-8. |
[18] |
A. Raich,
Compactness of the complex Green operator on CR-manifolds of hypersurface type, Math. Ann., 348 (2010), 81-117.
doi: 10.1007/s00208-009-0470-1. |
[19] |
A. Raich and E. Straube,
Compactness of the complex Green operator, Math. Res. Lett., 15 (2008), 761-778.
doi: 10.4310/MRL.2008.v15.n4.a13. |
[20] |
E. Straube, Lectures on the ${\mathcal{L}}^2$-Sobolev Theory of the $\bar\partial$-Neumann Problem, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2010.
doi: 10.4171/076. |
[21] |
E. J. Straube,
The complex Green operator on CR-submanifolds of $\mathbb{C}^n$ of hypersurface type: compactness, Trans. Amer. Math. Soc., 364 (2012), 4107-4125.
doi: 10.1090/S0002-9947-2012-05510-3. |
show all references
References:
[1] |
R. Basener,
Nonlinear Cauchy-Riemann equations and $q$-pseudoconvexity, Duke Math. J., 43 (1976), 203-213.
|
[2] |
S. Biard and E. Straube, $L^2$-Sobolev theory for the complex Green operator, Internat. J. Math., 28 (2017), 1740006, 31.
doi: 10.1142/S0129167X17400067. |
[3] |
A. Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, Florida, 1991. |
[4] |
D. Catlin, Global regularity of the $\bar\partial$-Neumann problem, in Complex analysis of several variables (Madison, Wis., 1982), Proc. Sympos. Pure Math., 41, Amer. Math. Soc., Providence, RI, 1984, 39-49.
doi: 10.1090/pspum/041/740870. |
[5] |
S. C. Chen and M. C. Shaw, Partial Differential Equations in Several Complex Variables, vol. 19 of Studies in Advanced Mathematics, American Mathematical Society, 2001.
doi: 10.11650/twjm/1500405913. |
[6] |
J. Coacalle and A. Raich,
Closed range estimates for $\bar\partial_b$ on CR manifolds of hypersurface type, J. Geom. Anal., 31 (2021), 366-394.
doi: 10.1007/s12220-019-00268-2. |
[7] |
R. Diaz,
Necessary conditions for subellipticity of ${\Box}_b$ on pseudoconvex domains, Commun. Partial Differ. Equ., 11 (1986), 1-61.
doi: 10.1080/03605308608820417. |
[8] |
R. Diaz,
Necessary conditions for local subellipticity of $\square_b$ on CR manifolds, J. Differ. Geom., 29 (1989), 389-419.
|
[9] |
G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann Complex, vol. 75 of Ann. of Math. Stud., Princeton University Press, Princeton, New Jersey, 1972. |
[10] |
P. Harrington and A. Raich,
Regularity results for $\bar\partial_b$ on CR-manifolds of hypersurface type, Commun. Partial Differ. Equ., 36 (2011), 134-161.
doi: 10.1080/03605302.2010.498855. |
[11] |
P. Harrington and A. Raich,
Closed range for $\bar\partial$ and $\bar\partial_b$ on bounded hypersurfaces in Stein manifolds, Ann. Inst. Fourier (Grenoble), 65 (2015), 1711-1754.
|
[12] |
P. S. Harrington, M. Peloso and A. Raich,
Regularity equivalence of the Szegö projection and the complex Green operator, Proc. Amer. Math. Soc., 143 (2015), 353-367.
doi: 10.1090/S0002-9939-2014-12393-8. |
[13] |
P. S. Harrington and A. Raich,
Closed range of $ \bar\partial$ in $L^2$-Sobolev spaces on unbounded domains in $ \mathbb C^n$, J. Math. Anal. Appl., 459 (2018), 1040-1461.
doi: 10.1016/j.jmaa.2017.11.017. |
[14] |
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
doi: 10.1017/CBO9780511810817.![]() ![]() ![]() |
[15] |
T. Khanh, S. Pinton and G. Zampieri,
Compactness estimates for $\square_{b}$ on a CR manifold, Proc. Amer. Math. Soc., 140 (2012), 3229-3236.
|
[16] |
K. Koenig,
A parametrix for the $\overline\partial$-Neumann problem on pseudoconvex domains of finite type, J. Funct. Anal., 216 (2004), 243-302.
doi: 10.1016/j.jfa.2004.06.004. |
[17] |
S. Munasinghe and E. Straube,
Geometric sufficient conditions for compactness of the complex Green operator, J. Geom. Anal., 22 (2012), 1007-1026.
doi: 10.1007/s12220-011-9226-8. |
[18] |
A. Raich,
Compactness of the complex Green operator on CR-manifolds of hypersurface type, Math. Ann., 348 (2010), 81-117.
doi: 10.1007/s00208-009-0470-1. |
[19] |
A. Raich and E. Straube,
Compactness of the complex Green operator, Math. Res. Lett., 15 (2008), 761-778.
doi: 10.4310/MRL.2008.v15.n4.a13. |
[20] |
E. Straube, Lectures on the ${\mathcal{L}}^2$-Sobolev Theory of the $\bar\partial$-Neumann Problem, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2010.
doi: 10.4171/076. |
[21] |
E. J. Straube,
The complex Green operator on CR-submanifolds of $\mathbb{C}^n$ of hypersurface type: compactness, Trans. Amer. Math. Soc., 364 (2012), 4107-4125.
doi: 10.1090/S0002-9947-2012-05510-3. |
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