Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The Cauchy transform on bounded domains


Authors: J. M. Anderson and A. Hinkkanen
Journal: Proc. Amer. Math. Soc. 107 (1989), 179-185
MSC: Primary 30E20; Secondary 47G05
DOI: https://doi.org/10.1090/S0002-9939-1989-0972226-5
MathSciNet review: 972226
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that $ f$ is in $ {L^2}(\Delta )$ where $ \Delta $ is the unit disk, and that $ f = 0$ outside $ \Delta $. We show that then the Cauchy transform $ \mathcal{C}\,f$ of $ f$, when restricted to $ \Delta $, satisfies $ \vert\vert\mathcal{C}\,f\vert{\vert _2} \leq (2/\alpha )\vert\vert f\vert{\vert _2}$, where $ \alpha \approx 2.4048$ is the smallest positive zero of the Bessel function $ {J_0}$. This inequality is sharp.


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

  • [1] D. W. Boyd, Best constants in a class of integral inequalities, Pacific J. Math. 30 (1969), 367-383. MR 0249556 (40:2801)
  • [2] D. H. Hamilton, On the Poincaré inequality, Complex Variables 5 (1986), 265-270. MR 846494 (87k:30003)
  • [3] O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, Springer, Berlin-Heidelberg-New York, 1973. MR 0344463 (49:9202)
  • [4] V. G. Maz'ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer, Berlin-Heidelberg-New York, 1985. MR 817985 (87g:46056)
  • [5] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, N.J., 1970. MR 0290095 (44:7280)
  • [6] G. N. Watson, A treatise on the theory of Bessel functions, 2nd edition, Cambridge University Press, Cambridge, 1962. MR 0010746 (6:64a)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30E20, 47G05

Retrieve articles in all journals with MSC: 30E20, 47G05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0972226-5
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society