$L^2$ estimates and existence theorems for $\protect \overline{\partial }_b$ on Lipschitz boundaries of $Q$-pseudoconvex domains

Sayed Saber

doi:10.5802/crmath.43

Analyse complexe

L^{2}

estimates and existence theorems for

{\bar{\partial}}_{b}

on Lipschitz boundaries of

Q

-pseudoconvex domains

Sayed Saber ¹

¹ Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Egypt

Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 435-458.

Résumé

On a bounded $q$ -pseudoconvex domain $Ω$ in $ℂ^{n}$ with Lipschitz boundary $b Ω$ , we prove the $L^{2}$ existence theorems of the ${\bar{\partial}}_{b}$ -operator on $b Ω$ . This yields the closed range property of ${\bar{\partial}}_{b}$ and its adjoint ${\bar{\partial}}_{b}^{*}$ . As an application, we establish the $L^{2}$ -existence theorems and regularity theorems for the ${\bar{\partial}}_{b}$ -Neumann operator.

Reçu le : 2019-05-01
Accepté le : 2020-03-24
Publié le : 2020-07-28

DOI : 10.5802/crmath.43

Classification : 35J20, 35J25, 35J60

Affiliations des auteurs :

Sayed Saber ¹

¹ Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Egypt

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Sayed Saber},
     title = {$L^2$ estimates and existence theorems for $\protect \overline{\partial }_b$ on {Lipschitz} boundaries of $Q$-pseudoconvex domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {435--458},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {4},
     year = {2020},
     doi = {10.5802/crmath.43},
     language = {en},
}

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Sayed Saber. $L^2$ estimates and existence theorems for $\protect \overline{\partial }_b$ on Lipschitz boundaries of $Q$-pseudoconvex domains. Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 435-458. doi : 10.5802/crmath.43. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.43/

Bibliographie
Cité par

[1] Heungju Ahn; Nguyen Quang Dieu The Donnelly–Fefferman theorem on $q$ -pseudoconvex domains, Osaka J. Math., Volume 46 (2009) no. 3, pp. 599-610 | MR | Zbl

[2] Aldo Andreotti; C. Denson Hill E. E. Levi convexity and the Hans Lewy problem. Parts I and II, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser., Volume 26 (1972), p. 325-363, 747–806 | Numdam | Zbl

[3] Harold P. Boas; Mei-Chi Shaw Sobolev estimates for the Lewy Operator on weakly pseudoconvex boundaries, Math. Ann., Volume 274 (1986), pp. 221-231 | DOI | MR | Zbl

[4] Harold P. Boas; Emil J. Straube Global regularity of the $\bar{\partial}$ -Neumann problem: A Survey of the $L^{2}$ -Sobolev Theory, Several complex variables (Mathematical Sciences Research Institute Publications), Volume 37 (1999), pp. 79-111 | MR | Zbl

[5] Albert Boggess CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Mathematics, CRC Press, 1991 | MR | Zbl

[6] So-Chin Chen; Mei-Chi Shaw Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, 19, American Mathematical Society, 2001 | MR | Zbl

[7] Lawrence C. Evans; Ronald F. Gariepy Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, 1992 | Zbl

[8] Gerald B. Folland; Elias M. Stein Estimates for the ${\bar{\partial}}_{b}$ complex and analysis on the Heisenberg group, Commun. Pure Appl. Math., Volume 27 (1974), pp. 429-522 | DOI | MR | Zbl

[9] Kurt O. Friedrichs The identity of weak and strong extensions of differential operators, Trans. Am. Math. Soc., Volume 55 (1944), pp. 132-151 | DOI | MR | Zbl

[10] Robert E. Greene; Hung-Hsi Wu Embedding of open Riemannian manifolds by harmonic functions, Ann. Inst. Fourier, Volume 25 (1975) no. 1, pp. 215-235 | DOI | Numdam | MR | Zbl

[11] Pierre Grisvard Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, 24, Pitman Publishing Inc., 1985 | MR | Zbl

[12] Le Mau Hai; Nguyen Quang Dieu; Nguyen Xuan Hong $L^{2}$ -Approximation of differential forms by $\bar{\partial}$ -closed ones on smooth hypersurfaces, J. Math. Anal. Appl., Volume 383 (2011) no. 2, pp. 379-390 | MR | Zbl

[13] Phillip S. Harrington Compactness and subellipticity for the $\bar{\partial}$ -Neumann operator on domains with minimal smoothness, Ph. D. Thesis, University of Notre Dame (USA) (2004) | MR

[14] Phillip S. Harrington; Andrew S. Raich Regularity results for ${\bar{\partial}}_{b}$ on CR-manifolds of hypersurface type, Commun. Partial Differ. Equations, Volume 36 (2011) no. 1-3, pp. 134-161 | DOI | MR | Zbl

[15] Phillip S. Harrington; Andrew S. Raich Closed range for $\bar{\partial}$ and ${\bar{\partial}}_{b}$ on bounded hypersurfaces in Stein manifolds, Ann. Inst. Fourier, Volume 65 (2015) no. 4, pp. 1711-1754 | DOI | Numdam | MR | Zbl

[16] Phillip S. Harrington; Andrew S. Raich Closed range of $\bar{\partial}$ in $L^{2}$ -Sobolev spaces on unbounded domains in $ℂ^{n}$ , J. Math. Anal. Appl., Volume 459 (2018) no. 2, pp. 1040-1061 | DOI | MR | Zbl

[17] Reese Harvey; John Polking Fundamental solutions in complex analysis, I and II, Duke Math. J., Volume 46 (1979), p. 253-300, 301–340 | DOI | Zbl

[18] Gennadi M. Henkin Solution des équations de Cauchy–Riemann tangentielles sur des variétés de Cauchy–Riemann $q$ -concaves, C. R. Math. Acad. Sci. Paris, Volume 292 (1981) no. 1, pp. 27-30 | Zbl

[19] Gennadi M. Henkin; Juergen Leiterer Global integral formulas for solving the $\bar{\partial}$ -equation on Stein manifolds, Ann. Pol. Math., Volume 39 (1981), pp. 93-116 | DOI | MR | Zbl

[20] Lop-Hing Ho $\bar{\partial}$ -problem on weakly $q$ -pseudoconvex domains, Math. Ann., Volume 290 (1991) no. 1, pp. 3-18 | MR | Zbl

[21] Lars Hörmander $L^{2}$ estimates and existence theorems for the $\bar{\partial}$ operator, Acta Math., Volume 113 (1965), pp. 89-152 | DOI | MR | Zbl

[22] David Jerison; Carlos E. Kenig The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., Volume 130 (1995) no. 1, pp. 161-219 | DOI | MR | Zbl

[23] Joseph J. Kohn Harmonic integrals on strongly pseudo-convex manifolds, I, Ann. Math., Volume 78 (1963), pp. 112-148 | DOI | MR | Zbl

[24] Joseph J. Kohn Global regularity for $\bar{\partial}$ on weakly pseudo-convex manifolds, Trans. Am. Math. Soc., Volume 181 (1973), pp. 273-292 | MR | Zbl

[25] Joseph J. Kohn The range of the tangential Cauchy–Riemann operator, Duke Math. J., Volume 53 (1986), pp. 525-545 | DOI | MR | Zbl

[26] Joseph J. Kohn; Hugo Rossi On the extension of holomorphic functions from the boundary of a complex manifold, Ann. Math., Volume 81 (1965), pp. 451-472 | DOI | MR | Zbl

[27] Christine Laurent-Thiébaut Sur l’équation de Cauchy–Riemann tangentielle dans une calotte strictement pseudoconvexe, Int. J. Math., Volume 16 (2005) no. 9, pp. 1063-1079 | DOI | MR | Zbl

[28] Jacques-Louis Lions; Enrico Magenes Non-homogeneous boundary value problems and applications I, Grundlehren der Mathematischen Wissenschaften, 181, Springer, 1972 | MR | Zbl

[29] Joachim Michel; Mei-Chi Shaw Subelliptic estimates for the $\bar{\partial}$ -Neumann operator on piecewise smooth strictly pseudoconvex domains, Duke Math. J., Volume 93 (1998) no. 1, pp. 115-128 | DOI | MR | Zbl

[30] Joachim Michel; Mei-Chi Shaw The $\bar{\partial}$ -Neumann operator on Lipschitz pseudoconvex domains with plurisubharmonic defining functions, Duke Math. J., Volume 108 (2001) no. 3, pp. 421-447 | DOI | MR | Zbl

[31] Andreea C. Nicoara Global regularity for ${\bar{\partial}}_{b}$ on weakly pseudoconvex CR manifolds, Adv. Math., Volume 199 (2006) no. 2, pp. 356-447 | DOI | MR | Zbl

[32] Jean Pierre Rosay Équation de Lewy-résolubilité globale de l’équation ${\bar{\partial}}_{b} u = f$ sur la frontiére de domaines faiblement pseudo-convexes de $C^{2}$ (ou $ℂ^{n}$ ), Duke Math. J., Volume 49 (1982) no. 1, pp. 121-128 | Zbl

[33] Mei-Chi Shaw Global solvability and regularity for $\bar{\partial}$ on an annulus between two weakly pseudoconvex domains, Trans. Am. Math. Soc., Volume 291 (1985), pp. 255-267 | MR | Zbl

[34] Mei-Chi Shaw $L^{2}$ estimates and existence theorems for the tangential Cauchy–Riemann complex, Invent. Math., Volume 82 (1985) no. 1, pp. 133-150 | DOI | MR | Zbl

[35] Mei-Chi Shaw $L^{2}$ existence theorems for the ${\bar{\partial}}_{b}$ -Neumann problem on strongly pseudoconvex CR manifolds, J. Geom. Anal., Volume 1 (1991) no. 2, pp. 139-163 | DOI | MR | Zbl

[36] Mei-Chi Shaw Local existence theorems with estimates for ${\bar{\partial}}_{b}$ on weakly pseudo-convex boundaries, Math. Ann., Volume 294 (1992) no. 4, pp. 677-700 | DOI | Zbl

[37] Mei-Chi Shaw $L^{2}$ estimates and existence theorems for ${\bar{\partial}}_{b}$ on Lipschitz boundaries, Math. Z., Volume 244 (2003) no. 1, pp. 91-123 | DOI | MR | Zbl

[38] Elias M. Stein Singular integrals and differentiability properties of functions, Princeton Mathematical Series, 30, Princeton University Press, 1970 | MR | Zbl

[39] Dennis Sullivan Hyperbolic geometry and homeomorphisms, Geometric topology (Athens, Georgia, 1977) (1979), pp. 543-555 | DOI | Zbl

[40] Nicolae Teleman The index of signature operators on Lipschitz manifolds, Publ. Math., Inst. Hautes Étud. Sci., Volume 58 (1983), pp. 39-78 | DOI | Numdam | MR | Zbl

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