Abstract
In this article, we establish a general sufficient condition for closed range of the Cauchy-Riemann operator \(\overline \partial\) in appropriately weighted L2 spaces on (0, q)-forms for a fixed q on domains in ℂn. The domains we consider may be neither bounded nor pseudoconvex, and our condition is a generalization of the classical Z(q) condition that we call weak Z(q). We provide examples that explain the necessity of working in weighted spaces for closed range in L2.
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References
H. Ahn, L. Baracco and G. Zampieri, Non-subelliptic estimates for the tangential Cauchy-Riemann system, Manuscripta Math. 121 (2006), 461–179.
A. Andreotti and H. Grauert, Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193–259.
S.-C. Chen and M.-C. Shaw, Partial Differential Equations in Several Complex Variables, American Mathematical Society, Providence, RI, 2001.
H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418–491.
G. B. Folland and J. J. Kohn, The Neumann Problem for the Cauchy-Riemann Complex, Princeton University Press, Princeton, NJ, 1972.
K. Gansberger, On the weighted \(\overline \partial \)-Neumann problem on unbounded domains, arXiv:0912.0841v1.
A.-K. Herbig and J. D. McNeal On closed range for \(\overline \partial \), Complex Var. Elliptic Equ. 61 (2016), 1073–1089.
L.-H. Ho, \(\overline \partial \) -problem on weakly q-convex domains, Math. Ann. 290 (1991), 3–18.
L. Hörmander, L 2 estimates and existence theorems for the \(\overline \partial \) operator, Acta Math. 113 (1965), 89–152.
P. Harrington and A. Raich, Regularity results for \({\overline \partial _b}\) on CR-manifolds of hypersurface type, Comm. Partial Differential Equations 36 (2011), 134–161.
P. Harrington and A. Raich, Defining functions for unbounded C m domains, Rev. Mat. Iberoam. 29 (2013), 1405–1420.
P. Harrington and A. Raich, Closed range for \(\overline \partial \) and \({\overline \partial _b}\) on bounded hypersurfaces in Stein manifolds, Ann. Inst. Fourier (Grenoble), 65 (2015), 1711–1754.
P. S. Harrington and A. Raich, A remark on boundary estimates on unbounded Z(q) domains in ℂ n, Complex Var. Elliptic Equ. 62 (2017), 1192–1203.
E. Straube, Lectures on the ℒ 2-Sobolev Theory of the \(\overline \partial \)-Neumann Problem, European Mathematical Society (EMS), Zürich, 2010.
G. Zampieri, Complex Analysis and CR Geometry, American Mathematical Society, Providence, RI, 2008.
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The second author was partially supported by NSF grant DMS-1405100.
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Harrington, P.S., Raich, A. Closed range of \(\overline \partial \) on unbounded domains in ℂn. JAMA 138, 185–208 (2019). https://doi.org/10.1007/s11854-019-0025-7
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DOI: https://doi.org/10.1007/s11854-019-0025-7