Advertisement

Mathematische Annalen

, Volume 374, Issue 1–2, pp 841–880 | Cite as

Hölder estimates for homotopy operators on strictly pseudoconvex domains with \(C^2\) boundary

  • Xianghong GongEmail author
Article
  • 279 Downloads

Abstract

We derive a new homotopy formula for a strictly pseudoconvex domain of \(C^2\) boundary in \({\mathbf C}^n\) by using a method of Lieb and Range and obtain estimates in Lipschitz spaces for the homotopy operators. For \(r>1\) and \(q>0\), we obtain a \(\Lambda _{r+{1}/{2}}\) solution u to \(\overline{\partial }u=f\) for a \(\overline{\partial }\)-closed (0, q)-form f of class \(\Lambda _{r}\) in the domain. We apply the estimates to obtain boundary regularities of \(\mathcal D\)-solutions for a domain in \(\mathbf{C}^n\times \mathbf{R}^m\).

Keyword

Mathematics Subject Classification

32A06 32T15 32W05 

Notes

Acknowledgements

The author is grateful to Andreas Seeger for helpful discussions on the real interpolation theory. The author would like to thank the referee for pointing out the precise references on the \(C^{1/2}\) estimates in the literature.

References

  1. 1.
    Ahern, P., Schneider, R.: Holomorphic Lipschitz functions in pseudoconvex domains. Am. J. Math. 101(3), 543–565 (1979).  https://doi.org/10.2307/2373797 MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alexandre, W.: \(C^k\)-estimates for the \(\overline{\partial }\)-equation on convex domains of finite type. Math. Z. 252(3), 473–496 (2006).  https://doi.org/10.1007/s00209-005-0812-y MathSciNetzbMATHGoogle Scholar
  3. 3.
    Alt, W.: Hölderabschätzungen für Ableitungen von Lösungen der Gleichung \(\bar{\partial }u=f\) bei streng pseudokonvexem Rand. Manuscr. Math. 13, 381–414 (1974)zbMATHGoogle Scholar
  4. 4.
    Brinkmann, Ch.: Lösungsoperatoren für den Cauchy-Riemann-Komplex auf Gebieten mit stückweise glattem, streng pseudokonvexem Rand in allgemeiner Lage mit \(C^k\)-Abschäitzungen, Diplomarbeit, Bonn, pp. 1–157 (1984)Google Scholar
  5. 5.
    Butzer, P.L., Berens, H.: Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band, vol. 145. Springer, New York (1967)Google Scholar
  6. 6.
    Calderón, A.-P.: Lebesgue spaces of differentiable functions and distributions. In: Proc. Sympos. Pure Math., vol. IV, pp. 33–49. American Mathematical Society, Providence (1961)Google Scholar
  7. 7.
    Catlin, D.: Subelliptic estimates for the \(\overline{\partial }\)-Neumann problem on pseudoconvex domains. Ann. Math. (2) 126(1), 131–191 (1987).  https://doi.org/10.2307/1971347 MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chang, D.-C.E.: Optimal \(L^p\) and Hölder estimates for the Kohn solution of the \(\overline{\partial }\)-equation on strongly pseudoconvex domains. Trans. Am. Math. Soc. 315(1), 273–304 (1989).  https://doi.org/10.2307/2001384 zbMATHGoogle Scholar
  9. 9.
    Chang, D.-C., Nagel, A., Stein, E.M.: Estimates for the \(\overline{\partial }\)-Neumann problem in pseudoconvex domains of finite type in \({\bf C}^2\). Acta Math. 169(3–4), 153–228 (1992).  https://doi.org/10.1007/BF02392760 MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chaumat, J., Chollet, A.-M.: Estimations höldériennes pour les équations de Cauchy-Riemann dans les convexes compacts de \({\bf C}^n\). Math. Z. 207(4), 501–534 (1991).  https://doi.org/10.1007/BF02571405 MathSciNetzbMATHGoogle Scholar
  11. 11.
    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 (2001)Google Scholar
  12. 12.
    Christ, M.: Regularity properties of the \(\overline{\partial }_b\) equation on weakly pseudoconvex CR manifolds of dimension 3. J. Am. Math. Soc. 1(3), 587–646 (1988).  https://doi.org/10.2307/1990950 zbMATHGoogle Scholar
  13. 13.
    Cumenge, A.: Sharp estimates for \(\overline{\partial }\) on convex domains of finite type. Ark. Mat. 39(1), 1–25 (2001).  https://doi.org/10.1007/BF02388789 MathSciNetzbMATHGoogle Scholar
  14. 14.
    D’Angelo, J.P.: Real hypersurfaces, orders of contact, and applications. Ann. Math. (2) 115(3), 615–637 (1982).  https://doi.org/10.2307/2007015 MathSciNetzbMATHGoogle Scholar
  15. 15.
    Diederich, K., Fischer, B., Fornæss, J.E.: Hölder estimates on convex domains of finite type. Math. Z. 232(1), 43–61 (1999).  https://doi.org/10.1007/PL00004758 MathSciNetzbMATHGoogle Scholar
  16. 16.
    Diederich, K., Fornæss, J.E., Wiegerinck, J.: Sharp Hölder estimates for \(\overline{\partial }\) on ellipsoids. Manuscr. Math. 56(4), 399–417 (1986).  https://doi.org/10.1007/BF01168502 zbMATHGoogle Scholar
  17. 17.
    Elgueta, M.: Extension to strictly pseudoconvex domains of functions holomorphic in a submanifold in general position and \(C^{\infty }\) up to the boundary. Ill. J. Math. 24(1), 1–17 (1980)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Fefferman, C.L., Kohn, J.J.: Hölder estimates on domains of complex dimension two and on three-dimensional CR manifolds. Adv. Math. 69(2), 223–303 (1988).  https://doi.org/10.1016/0001-8708(88)90002-3 MathSciNetzbMATHGoogle Scholar
  19. 19.
    Fefferman, C.L., Kohn, J.J., Machedon, M.: Hölder estimates on CR manifolds with a diagonalizable Levi form. Adv. Math. 84(1), 1–90 (1990).  https://doi.org/10.1016/0001-8708(90)90036-M MathSciNetzbMATHGoogle Scholar
  20. 20.
    Folland, G.B., Stein, E.M.: Estimates for the \(\bar{\partial }_{b}\) complex and analysis on the Heisenberg group. Commun. Pure Appl. Math. 27, 429–522 (1974)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Glaeser, G.: Étude de quelques algèbres tayloriennes. J. Anal. Math. 6, 1–124 (1958).  https://doi.org/10.1007/BF02790231
  22. 22.
    Gong, X.: A Frobenius–Nirenberg theorem with parameter. J. Reine Angew. Math.  https://doi.org/10.1515/crelle-2017-0051
  23. 23.
    Gong, X., Kim, K.-T.: The \(\overline{\partial }\)-equation on variable strictly pseudoconvex domains. Math. Z. (2017).  https://doi.org/10.1007/s00209-017-2011-z
  24. 24.
    Gong, X., Webster, S.M.: Regularity for the CR vector bundle problem II. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10(1), 129–191 (2011)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Grauert, H., Lieb, I.: Das Ramirezsche Integral und die Lösung der Gleichung \(\bar{\partial }f=\alpha \) im Bereich der beschränkten Formen. Rice Univ. Stud. 56(2), 29–50 (1970)zbMATHGoogle Scholar
  26. 26.
    Greiner, P.C., Stein, E.M.: Estimates for the \(\overline{\partial }\)-Neumann problem, Mathematical Notes, No. 19. Princeton University Press, Princeton (1977)Google Scholar
  27. 27.
    Hanges, N., Jacobowitz, H.: The Euclidean elliptic complex. Indiana Univ. Math. J. 46(3), 753–770 (1997).  https://doi.org/10.1512/iumj.1997.46.1397 MathSciNetzbMATHGoogle Scholar
  28. 28.
    Henkin, G.M.: Integral representation of functions which are holomorphic in strictly pseudoconvex regions, and some applications. Mat. Sb. (N.S.) 78(120), 611–632 (1969)MathSciNetGoogle Scholar
  29. 29.
    Henkin, G.M., Leiterer, J.: Theory of functions on complex manifolds, Monographs in Mathematics, vol. 79, p. 226. Birkhäuser, Basel (1984)Google Scholar
  30. 30.
    Henkin, G.M., Romanov, A.V.: Exact Hölder estimates of the solutions of the \(\bar{\delta }\)-equation. Izv. Akad. Nauk SSSR Ser. Mat. 35, 1171–1183 (1971)MathSciNetGoogle Scholar
  31. 31.
    Kerzman, N.: Hölder and \(L^{p}\) estimates for solutions of \(\bar{\partial }u=f\) in strongly pseudoconvex domains. Commun. Pure Appl. Math. 24, 301–379 (1971)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Koenig, K.D.: On maximal Sobolev and Hölder estimates for the tangential Cauchy–Riemann operator and boundary Laplacian. Am. J. Math. 124(1), 129–197 (2002)zbMATHGoogle Scholar
  33. 33.
    Kohn, J.J.: Harmonic integrals on strongly pseudo-convex manifolds. II. Ann. Math. (2) 79, 450–472 (1964)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Kohn, J.J.: Global regularity for \(\bar{\partial }\) on weakly pseudo-convex manifolds. Trans. Am. Math. Soc. 181, 273–292 (1973)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Lieb, I.: Die Cauchy–Riemannschen Differentialgleichungen auf streng pseudokonvexen Gebieten. Beschränkte Lösungen. Math. Ann. 190(1970), 6–44 (1971)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Lieb, I., Range, R.M.: Lösungsoperatoren für den Cauchy–Riemann–Komplex mit \({\cal{C}}^{k}\)-Abschätzungen. Math. Ann. 253(2), 145–164 (1980).  https://doi.org/10.1007/BF01578911 MathSciNetzbMATHGoogle Scholar
  37. 37.
    Lieb, I., Range, R.M.: Integral representations and estimates in the theory of the \(\overline{\partial }\)-Neumann problem. Ann. Math. (2) 123(2), 265–301 (1986).  https://doi.org/10.2307/1971272 MathSciNetzbMATHGoogle Scholar
  38. 38.
    Lieb, I., Range, R.M.: Estimates for a class of integral operators and applications to the \(\overline{\partial }\)-Neumann problem. Invent. Math. 85(2), 415–438 (1986).  https://doi.org/10.1007/BF01389097 MathSciNetzbMATHGoogle Scholar
  39. 39.
    Ma, L., Michel, J.: Local regularity for the tangential Cauchy–Riemann complex. J. Reine Angew. Math. 442, 63–90 (1993).  https://doi.org/10.1515/crll.1993.442.63 MathSciNetzbMATHGoogle Scholar
  40. 40.
    Michel, J.: Randregularität des \(\overline{\partial }\)-problems für stückweise streng pseudokonvexe Gebiete in \({\bf C}^n\), Math. Ann. 280(1), 45–68 (1988).  https://doi.org/10.1007/BF01474180 MathSciNetGoogle Scholar
  41. 41.
    Michel, J.: Integral representations on weakly pseudoconvex domains. Math. Z. 208(3), 437–462 (1991).  https://doi.org/10.1007/BF02571538 MathSciNetzbMATHGoogle Scholar
  42. 42.
    Michel, J., Perotti, A.: \(C^k\)-regularity for the \(\overline{\partial }\)-equation on strictly pseudoconvex domains with piecewise smooth boundaries. Math. Z. 203(3), 415–427 (1990).  https://doi.org/10.1007/BF02570747 MathSciNetzbMATHGoogle Scholar
  43. 43.
    Michel, J., Shaw, M.-C.: A decomposition problem on weakly pseudoconvex domains. Math. Z. 230(1), 1–19 (1999).  https://doi.org/10.1007/PL00004685 MathSciNetzbMATHGoogle Scholar
  44. 44.
    McNeal, J.D.: Estimates on the Bergman kernels of convex domains. Adv. Math. 109(1), 108–139 (1994)MathSciNetzbMATHGoogle Scholar
  45. 45.
    McNeal, J.D., Stein, E.M.: Mapping properties of the Bergman projection on convex domains of finite type. Duke Math. J. 73(1), 177–199 (1994).  https://doi.org/10.1215/S0012-7094-94-07307-9 MathSciNetzbMATHGoogle Scholar
  46. 46.
    Øvrelid, N.: Integral representation formulas and \(L^{p}\)-estimates for the \(\bar{\partial }\)-equation. Math. Scand. 29, 137–160 (1971)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Peters, K.: Solution operators for the \(\overline{\partial }\)-equation on nontransversal intersections of strictly pseudoconvex domains. Math. Ann. 291(4), 617–641 (1991).  https://doi.org/10.1007/BF01445231 MathSciNetzbMATHGoogle Scholar
  48. 48.
    Phong, D.H., Stein, E.M.: Estimates for the Bergman and Szegö projections on strongly pseudo-convex domains. Duke Math. J. 44(3), 695–704 (1977)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Poljakov, P.L.: Banach cohomology on piecewise strictly pseudoconvex domains. Mat. Sb. (N.S.) 88(130), 238–255 (1972)MathSciNetGoogle Scholar
  50. 50.
    de Arellano, Ramírez: E.: Ein Divisionsproblem und Randintegraldarstellungen in der komplexen Analysis. Math. Ann. 184(1969), 172–187 (1970)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Range, R.M.: Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, vol. 108, Springer, New York (1986)Google Scholar
  52. 52.
    Range, R.M.: Integral kernels and Hölder estimates for \(\overline{\partial }\) on pseudoconvex domains of finite type in \({\bf C}^2\). Math. Ann. 288(1), 63–74 (1990).  https://doi.org/10.1007/BF01444521 MathSciNetzbMATHGoogle Scholar
  53. 53.
    Range, R.M., Siu, Y.-T.: Uniform estimates for the \(\bar{\partial }\)-equation on domains with piecewise smooth strictly pseudoconvex boundaries. Math. Ann. 206, 325–354 (1973).  https://doi.org/10.1007/BF01355986 MathSciNetzbMATHGoogle Scholar
  54. 54.
    Shaw, M.-C.: Optimal Hölder and \(L^p\) estimates for \(\overline{\partial }_b\) on the boundaries of real ellipsoids in \({\bf C}^n\). Trans. Am. Math. Soc. 324(1), 213–234 (1991).  https://doi.org/10.2307/2001504 Google Scholar
  55. 55.
    Siu, Y.-T.: The \(\bar{\partial }\) problem with uniform bounds on derivatives. Math. Ann. 207, 163–176 (1974)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Spivak, M.: A comprehensive introduction to differential geometry, vol. I, 3rd edn. Publish or Perish, Inc., Houston (1999)Google Scholar
  57. 57.
    Stein, E.M.: Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton (1970)Google Scholar
  58. 58.
    Treves, F.: Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Local theory. Princeton University Press, Princeton (1992)Google Scholar
  59. 59.
    Triebel, H.: Interpolation theory, function spaces, differential operators, 2nd ed. Johann Ambrosius Barth, Heidelberg (1995)Google Scholar
  60. 60.
    Webster, S.M.: A new proof of the Newlander–Nirenberg theorem. Math. Z. 201(3), 303–316 (1989).  https://doi.org/10.1007/BF01214897 MathSciNetzbMATHGoogle Scholar
  61. 61.
    Webster, S.M.: On the local solution of the tangential Cauchy–Riemann equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 6(3), 167–182 (1989)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Whitney, H.: Analytic extensions of differentiable functions defined in closed sets. Trans. Am. Math. Soc. 36(1), 63–89 (1934).  https://doi.org/10.2307/1989708 MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA

Personalised recommendations