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Supremum estimates for ∂ˉ\bar\partial on bounded convex domains in C2\mathbb{C}^2

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Contributed by Yun-Heng Du ↗

Version 1 / Nov 05, 2026 / CC BY 4.0

Abstract

For every bounded convex domain Ω⊂C2\Omega\subset\mathbb{C}^2, we give a bounded distributional solution of ∂ˉu=f\bar\partial u=f for every distributionally ∂ˉ\bar\partial-closed form f∈L(0,1)∞(Ω)f\in L^\infty_{(0,1)}(\Omega). The solution satisfies ∥u∥∞≤CΩ∥f∥∞\|u\|_\infty\le C_\Omega\|f\|_\infty, where CΩ>0C_\Omega>0 depends only on Ω\Omega. The estimate follows from holomorphic localization of Leray rows and boundary cancellation.

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References

  1. L. Alaoglu, Weak topologies of normed linear spaces , Ann. of Math. (2) 41 (1940), 252--267. doi:10.2307/1968829 (https://doi.org/10.2307/1968829).DOI
  2. B. Berndtsson, A smooth pseudoconvex domain in $\C^2$ for which L∞L^\infty-estimates for $\dbar$ do not hold , Ark. Mat. 31 (1993), 209--218. doi:10.1007/BF02559484 (https://doi.org/10.1007/BF02559484).DOI
  3. A. Cumenge, Sharp estimates for $\dbar$ on convex domains of finite type , Ark. Mat. 39 (2001), 1--25. doi:10.1007/BF02388789 (https://doi.org/10.1007/BF02388789).DOI
  4. K. Diederich, B. Fischer, and J. E. Fornæss, Hölder estimates on convex domains of finite type , Math. Z. 232 (1999), 43--61. doi:10.1007/PL00004758 (https://doi.org/10.1007/PL00004758).DOI
  5. C. L. Fefferman and J. J. Kohn, Hölder estimates on domains of complex dimension two and on three dimensional CR manifolds , Adv. Math. 69 (1988), 223--303. doi:10.1016/0001-8708(88)90002-3 (https://doi.org/10.1016/0001-8708(88)90002-3).DOI
  6. G. B. Folland, Real Analysis: Modern Techniques and Their Applications , 2nd ed., John Wiley & Sons, New York, 1999.
  7. J. E. Fornæss and K.-T. Kim, Some problems , in Complex Analysis and Geometry, KSCV10, Gyeongju, Korea, August 2014 , Springer Proc. Math. Stat. 144 , Springer, Tokyo, 2015, 369--377. doi:10.1007/978-4-431-55744-9_28 (https://doi.org/10.1007/978-4-431-55744-9_28).DOI
  8. J. E. Fornæss, L. Lee, and Y. Zhang, On supnorm estimates for $\dbar$ on infinite type convex domains in $\C^2$ , J. Geom. Anal. 21 (2011), 495--512. doi:10.1007/s12220-010-9155-y (https://doi.org/10.1007/s12220-010-9155-y).DOI
  9. J. E. Fornæss, L. Lee, and Y. Zhang, On supnorm estimates for $\dbar$ on infinite type convex domains in $\C^2$ , preprint, 12 November 2009, arXiv:0911.2277v1 (https://arxiv.org/abs/0911.2277v1).arXiv
  10. X. Gong, Hölder estimates for homotopy operators on strictly pseudoconvex domains with C2C^2 boundary , Math. Ann. 374 (2019), 841--880. doi:10.1007/s00208-018-1693-9 (https://doi.org/10.1007/s00208-018-1693-9).DOI
Show all 21 referencesShow fewer references
  1. H. Grauert and I. Lieb, Das Ramirezsche Integral und die Lösung der Gleichung $\dbar f=\alpha$ im Bereich der beschränkten Formen , Rice Univ. Stud. 56 (1970), no. 2, 29--50.
  2. H. Hefer, Zur Funktionentheorie mehrerer Veränderlichen. Über eine Zerlegung analytischer Funktionen und die Weilsche Integraldarstellung , Math. Ann. 122 (1950), 276--278. doi:10.1007/BF01342970 (https://doi.org/10.1007/BF01342970).DOI
  3. G. M. Henkin, Integral representation of functions in strictly pseudoconvex domains and applications to the $\dbar$-problem , Math. USSR-Sb. 11 (1970), 273--281. doi:10.1070/SM1970v011n02ABEH002069 (https://doi.org/10.1070/SM1970v011n02ABEH002069).DOI
  4. G. M. Henkin, A uniform estimate for the solution of the $\dbar$-problem in a Weil region (Russian), Uspekhi Mat. Nauk 26 (1971), no. 3, 211--212.
  5. T. V. Khanh, Supnorm and ff-Hölder estimates for $\dbar$ on convex domains of general type in $\C^2$ , J. Math. Anal. Appl. 403 (2013), 522--531. doi:10.1016/j.jmaa.2013.02.045 (https://doi.org/10.1016/j.jmaa.2013.02.045).DOI
  6. W. Koppelman, The Cauchy integral for differential forms , Bull. Amer. Math. Soc. 73 (1967), 554--556. doi:10.1090/S0002-9904-1967-11744-0 (https://doi.org/10.1090/S0002-9904-1967-11744-0).DOI
  7. R. M. Range, Hölder estimates for $\dbar$ on convex domains in $\C^2$ with real analytic boundary , in Several Complex Variables , Proc. Sympos. Pure Math. 30 , part 2, Amer. Math. Soc., Providence, RI, 1977, 31--33. doi:10.1090/pspum/030.2/0450619 (https://doi.org/10.1090/pspum/030.2/0450619).DOI
  8. R. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables , Grad. Texts in Math. 108 , Springer, New York, 1986. doi:10.1007/978-1-4757-1918-5 (https://doi.org/10.1007/978-1-4757-1918-5).DOI
  9. R. M. Range, On Hölder and BMO estimates for $\dbar$ on convex domains in $\C^2$ , J. Geom. Anal. 2 (1992), 575--584. doi:10.1007/BF02921578 (https://doi.org/10.1007/BF02921578).DOI
  10. R. M. Range, On $\dbar$-problems on (pseudo)-convex domains , in Topics in Complex Analysis , Banach Center Publ. 31 , Polish Acad. Sci., Warsaw, 1995, 311--320. doi:10.4064/-31-1-311-320 (https://doi.org/10.4064/-31-1-311-320).DOI
  11. Y. Yuan, Uniform estimates of the Cauchy--Riemann equation and Kerzman's question , Peking Math. J. 9 (2026), 629--642. doi:10.1007/s42543-025-00096-0 (https://doi.org/10.1007/s42543-025-00096-0).DOI

Version history

  1. v1 · 22 pagesSubmitted by Yun-Heng DuInitial depositCurrentNov 05, 2026