Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Estimates for the $ \bar \partial $-Neumann operator in weighted Hilbert spaces

Author: Sidney L. Hantler
Journal: Trans. Amer. Math. Soc. 217 (1976), 395-406
MSC: Primary 32A15; Secondary 30A82
MathSciNet review: 0393535
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Estimates for the $ \bar \partial $ operator are used to derive estimates for the Neumann operator in weighted Hilbert spaces. The technique is similar to that used to prove regularity of solutions of elliptic partial differential equations. A priori estimates are first obtained for smooth compactly supported forms and these estimates are then extended by suitable approximation results. These estimates are applied to give new bounds for the reproducing kernels in the subspaces of entire functions.

References [Enhancements On Off] (What's this?)

  • [1] Stefan Bergman, The Kernel Function and Conformal Mapping, Mathematical Surveys, No. 5, American Mathematical Society, New York, N. Y., 1950. MR 0038439
  • [2] S. L. Hantler, Estimates for reproducing kernels in weighted Hilbert spaces of entire functions, RC 4926, IBM T. J. Watson Research Center, Yorktown Heights, New York, 1974, 117 pp.
  • [3] -, Polynomial approximation in certain weighted Hilbert spaces of entire functions (to appear).
  • [4] Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
  • [5] Lars Hörmander, 𝐿² estimates and existence theorems for the ∂ operator, Acta Math. 113 (1965), 89–152. MR 0179443
  • [6] Norberto Kerzman, The Bergman kernel function. Differentiability at the boundary, Math. Ann. 195 (1972), 149–158. MR 0294694
  • [7] I. J. Kohn, Harmonic integrals on strongly pseudoconvex manifolds. I, II, Ann. of Math. (2) 78 (1963), 112-148; ibid. (2) 79 (1964), 450-472. MR 27 #2999; 34 #8010.
  • [8] D. J. Newman and H. S. Shapiro, A Hilbert space of entire functions related to the operational calculus, University of Michigan, Ann Arbor, Mich., 1964, 92 pp. (mimeographed notes).
  • [9] B. A. Taylor, On weighted polynomial approximation of entire functions, Pacific J. Math. 36 (1971), 523–539. MR 0284801

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32A15, 30A82

Retrieve articles in all journals with MSC: 32A15, 30A82

Additional Information

Keywords: $ \bar \partial $-Neumann problem, $ \bar \partial $ operator, weighted Hilbert spaces, reproducing kernel, Bergman kernel function
Article copyright: © Copyright 1976 American Mathematical Society