# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021061

## 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

* Corresponding author

Received  May 2020 Revised  February 2021 Published  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.

Citation: Joel Coacalle, Andrew Raich. Compactness of the complex Green operator on non-pseudoconvex CR manifolds. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021061
##### References:
 [1] R. Basener, Nonlinear Cauchy-Riemann equations and $q$-pseudoconvexity, Duke Math. J., 43 (1976), 203-213.   Google Scholar [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.  Google Scholar [3] A. Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, Florida, 1991.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] R. Diaz, Necessary conditions for local subellipticity of $\square_b$ on CR manifolds, J. Differ. Geom., 29 (1989), 389-419.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [14] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.  doi: 10.1017/CBO9780511810817.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### References:
 [1] R. Basener, Nonlinear Cauchy-Riemann equations and $q$-pseudoconvexity, Duke Math. J., 43 (1976), 203-213.   Google Scholar [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.  Google Scholar [3] A. Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, Florida, 1991.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] R. Diaz, Necessary conditions for local subellipticity of $\square_b$ on CR manifolds, J. Differ. Geom., 29 (1989), 389-419.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [14] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.  doi: 10.1017/CBO9780511810817.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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