• Previous Article
    Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics
  • CPAA Home
  • This Issue
  • Next Article
    On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation
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. HarringtonM. 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. KhanhS. 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

show all references

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. HarringtonM. 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. KhanhS. 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

[1]

Phuong Le. Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 511-525. doi: 10.3934/cpaa.2020025

[2]

C. David Levermore, Weiran Sun. Compactness of the gain parts of the linearized Boltzmann operator with weakly cutoff kernels. Kinetic & Related Models, 2010, 3 (2) : 335-351. doi: 10.3934/krm.2010.3.335

[3]

Joel Andersson, Leo Tzou. Stability for a magnetic Schrödinger operator on a Riemann surface with boundary. Inverse Problems & Imaging, 2018, 12 (1) : 1-28. doi: 10.3934/ipi.2018001

[4]

Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665

[5]

Saikat Mazumdar. Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 311-330. doi: 10.3934/cpaa.2017015

[6]

Alonso Sepúlveda, Guilherme Tizziotti. Weierstrass semigroup and codes over the curve $y^q + y = x^{q^r + 1}$. Advances in Mathematics of Communications, 2014, 8 (1) : 67-72. doi: 10.3934/amc.2014.8.67

[7]

Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311

[8]

Laurent Amour, Jérémy Faupin. Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials. Inverse Problems & Imaging, 2013, 7 (4) : 1115-1122. doi: 10.3934/ipi.2013.7.1115

[9]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko. A partial inverse problem for the Sturm-Liouville operator on the lasso-graph. Inverse Problems & Imaging, 2019, 13 (1) : 69-79. doi: 10.3934/ipi.2019004

[10]

Shingo Takeuchi. Partial flat core properties associated to the $p$-laplace operator. Conference Publications, 2007, 2007 (Special) : 965-973. doi: 10.3934/proc.2007.2007.965

[11]

Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169

[12]

Leyter Potenciano-Machado, Alberto Ruiz. Stability estimates for a magnetic Schrödinger operator with partial data. Inverse Problems & Imaging, 2018, 12 (6) : 1309-1342. doi: 10.3934/ipi.2018055

[13]

Horst R. Thieme. Positive perturbation of operator semigroups: growth bounds, essential compactness and asynchronous exponential growth. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 735-764. doi: 10.3934/dcds.1998.4.735

[14]

Benoît Pausader, Walter A. Strauss. Analyticity of the nonlinear scattering operator. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 617-626. doi: 10.3934/dcds.2009.25.617

[15]

Vittorio Martino. On the characteristic curvature operator. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1911-1922. doi: 10.3934/cpaa.2012.11.1911

[16]

Lijian Jiang, Craig C. Douglas. Analysis of an operator splitting method in 4D-Var. Conference Publications, 2009, 2009 (Special) : 394-403. doi: 10.3934/proc.2009.2009.394

[17]

Sombuddha Bhattacharyya. An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data. Inverse Problems & Imaging, 2018, 12 (3) : 801-830. doi: 10.3934/ipi.2018034

[18]

Kangkang Deng, Zheng Peng, Jianli Chen. Sparse probabilistic Boolean network problems: A partial proximal-type operator splitting method. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1881-1896. doi: 10.3934/jimo.2018127

[19]

Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040

[20]

Peter C. Gibson. On the measurement operator for scattering in layered media. Inverse Problems & Imaging, 2017, 11 (1) : 87-97. doi: 10.3934/ipi.2017005

2019 Impact Factor: 1.105

Article outline

[Back to Top]