The ∂ ¯ -Neumann operator on Lipschitz q -pseudoconvex domains

Sayed Saber

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 3, page 721-731
  • ISSN: 0011-4642

Abstract

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On a bounded q -pseudoconvex domain Ω in ℂ n with a Lipschitz boundary, we prove that the ∂ ¯ -Neumann operator N satisfies a subelliptic ( 1 / 2 ) -estimate on Ω and N can be extended as a bounded operator from Sobolev ( - 1 / 2 ) -spaces to Sobolev ( 1 / 2 ) -spaces.

How to cite

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Saber, Sayed. "The $\bar{\partial }$-Neumann operator on Lipschitz $q$-pseudoconvex domains." Czechoslovak Mathematical Journal 61.3 (2011): 721-731. <http://eudml.org/doc/197031>.

@article{Saber2011,
abstract = {On a bounded $q$-pseudoconvex domain $\Omega $ in $\mathbb \{C\}^\{n\}$ with a Lipschitz boundary, we prove that the $\bar\{\partial \}$-Neumann operator $N$ satisfies a subelliptic $(1/2)$-estimate on $\Omega $ and $N$ can be extended as a bounded operator from Sobolev $(-1/2)$-spaces to Sobolev $(1/2)$-spaces.},
author = {Saber, Sayed},
journal = {Czechoslovak Mathematical Journal},
keywords = {Sobolev estimate; $\bar\{\partial \}$ and $\bar\{\partial \}$-Neumann operator; $q$-pseudoconvex domains; Lipschitz domains; Sobolev estimate; operator; -Neumann operator; -pseudoconvex domain; Lipschitz domain},
language = {eng},
number = {3},
pages = {721-731},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The $\bar\{\partial \}$-Neumann operator on Lipschitz $q$-pseudoconvex domains},
url = {http://eudml.org/doc/197031},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Saber, Sayed
TI - The $\bar{\partial }$-Neumann operator on Lipschitz $q$-pseudoconvex domains
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 3
SP - 721
EP - 731
AB - On a bounded $q$-pseudoconvex domain $\Omega $ in $\mathbb {C}^{n}$ with a Lipschitz boundary, we prove that the $\bar{\partial }$-Neumann operator $N$ satisfies a subelliptic $(1/2)$-estimate on $\Omega $ and $N$ can be extended as a bounded operator from Sobolev $(-1/2)$-spaces to Sobolev $(1/2)$-spaces.
LA - eng
KW - Sobolev estimate; $\bar{\partial }$ and $\bar{\partial }$-Neumann operator; $q$-pseudoconvex domains; Lipschitz domains; Sobolev estimate; operator; -Neumann operator; -pseudoconvex domain; Lipschitz domain
UR - http://eudml.org/doc/197031
ER -

References

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  1. Abdelkader, O., Saber, S., Estimates for the ∂ ¯ -Neumann operator on strictly pseudo-convex domain with Lipschitz boundary, J. Inequal. Pure Appl. Math. 5 10 (2004). (2004) MR2084879
  2. Abdelkader, O., Saber, S., The ∂ ¯ -Neumann operator on a strictly pseudo-convex domain with piecewise smooth boundary, Math. Slovaca 55 (2005), 317-328. (2005) MR2181009
  3. Ahn, H., Dieu, N. Q., The Donnelly-Fefferman Theorem on q -pseudoconvex domains, Osaka J. Math. 46 (2009), 599-610. (2009) Zbl1214.32015MR2583320
  4. Boas, H. P., Straube, E. J., Global regularity of the ∂ ¯ -Neumann problem: A Survey of the L 2 -Sobolev Theory, Several Complex Variables, MSRI Publications 37 (1999), 79-111. (1999) MR1748601
  5. Boas, H. P., Straube, E. J., 10.1007/BF02571327, Math. Z. 206 (1991), 81-88. (1991) MR1086815DOI10.1007/BF02571327
  6. Bonami, A., Charpentier, P., Boundary values for the canonical solution to ∂ ¯ -equation and W 1 / 2 estimates, Preprint, Bordeaux (1990). (1990) MR1055987
  7. Catlin, D., 10.2307/1971347, Annals Math. 126 (1987), 131-191. (1987) Zbl0627.32013MR0898054DOI10.2307/1971347
  8. Chen, S. C., Shaw, M. C., Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI (2001). (2001) Zbl0963.32001MR1800297
  9. Ehsani, D., 10.4310/MRL.2003.v10.n4.a11, Math. Res. Letters 10 (2003), 523-533. (2003) MR1995791DOI10.4310/MRL.2003.v10.n4.a11
  10. Ehsani, D., 10.1512/iumj.2003.52.2261, Indiana Univ. Math. J. 52 (2003), 629-666. (2003) MR1986891DOI10.1512/iumj.2003.52.2261
  11. Engliš, M., 10.1512/iumj.2001.50.2085, Indiana Univ. Math. J. 50 (2001), 1593-1607. (2001) MR1889072DOI10.1512/iumj.2001.50.2085
  12. Evans, L. E., Gariepy, R. F., Measure theory and fine properties of functions, Stud. Adv. Math., CRC, Boca Raton (1992). (1992) Zbl0804.28001MR1158660
  13. Folland, G. B., Kohn, J. J., The Neumann problem for the Cauchy-Riemann complex, Ann. Math. Studies , Princeton University Press, New York, 1972. Zbl0247.35093MR0461588
  14. Grisvard, P., Elliptic problems in nonsmooth domains, Monogr. Stud. Math. Pitman, Boston 24 (1985). (1985) Zbl0695.35060MR0775683
  15. Henkin, G., Iordan, A., Kohn, J. J., Estimations sous-elliptiques pour le problème ∂ ¯ -Neumann dans un domaine strictement pseudoconvexe à frontière lisse par morceaux, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 17-22. (1996) MR1401622
  16. Ho, L.-H., 10.1007/BF01459235, Math. Ann. 290 (1991), 3-18. (1991) Zbl0714.32006MR1107660DOI10.1007/BF01459235
  17. Hörmander, L., 10.1007/BF02391775, Acta Math. 113 (1965), 89-152. (1965) MR0179443DOI10.1007/BF02391775
  18. Kohn, J. J., Global regularity for ∂ ¯ on weakly pseudo-convex manifolds, Trans. Amer. Math. Soc. 181 (1973), 273-292. (1973) Zbl0276.35071MR0344703
  19. Kohn, J. J., 10.2307/1970506, Ann. Math. 78 (1963), 112-148. (1963) MR0153030DOI10.2307/1970506
  20. Kohn, J. J., 10.2307/1970404, Ann. Math. 79 (1964), 450-472. (1964) MR0208200DOI10.2307/1970404
  21. Michel, J., Shaw, M., 10.1215/S0012-7094-98-09304-8, Duke Math. J. 93 (1998), 115-128. (1998) MR1620087DOI10.1215/S0012-7094-98-09304-8
  22. Michel, J., Shaw, M., 10.1215/S0012-7094-01-10832-6, Duke Math. J. 108 (2001), 421-447. (2001) MR1838658DOI10.1215/S0012-7094-01-10832-6
  23. Stein, E. M., Singular integrals and differentiability properties of functions, Princeton, Princeton Univ. Press Vol. 30 (1970). (1970) Zbl0207.13501MR0290095
  24. Straube, E., 10.4310/MRL.1997.v4.n4.a2, Math. Res. Lett. 4 (1997), 459-467. (1997) MR1470417DOI10.4310/MRL.1997.v4.n4.a2
  25. Zampieri, G., 10.1023/A:1001811318865, Compositio Math. 121 (2000), 155-162. (2000) MR1757879DOI10.1023/A:1001811318865

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