Compactness of the complex Green operator on non-pseudoconvex CR manifolds

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June 2021, 20(6): 2139-2154. doi: 10.3934/cpaa.2021061

Compactness of the complex Green operator on non-pseudoconvex CR manifolds

Joel Coacalle ^1, and Andrew Raich ^2,,

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

^* Corresponding author

Received May 2020 Revised February 2021 Published June 2021 Early access April 2021

Fund Project: This work was supported by a grant from the Simons Foundation (707123, ASR)

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.

Keywords: Weak $ Z(q) $, weak $ Y(q) $, tangential Cauchy-Riemann operator, $ \bar\partial_b $, closed range, microlocal analysis, complex Green operator, compactness, compact operator.

Mathematics Subject Classification: Primary: 32W10, Secondary: 32F17, 32V20, 35A27, 35N15.

Citation: Joel Coacalle, Andrew Raich. Compactness of the complex Green operator on non-pseudoconvex CR manifolds. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2139-2154. doi: 10.3934/cpaa.2021061

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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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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