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Some sharp weak-type inequalities for holomorphic functions on the unit ball of $ {\bf C}\sp n$


Author: Bogusław Tomaszewski
Journal: Proc. Amer. Math. Soc. 95 (1985), 271-274
MSC: Primary 32A35; Secondary 32A40
DOI: https://doi.org/10.1090/S0002-9939-1985-0801337-6
MathSciNet review: 801337
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Abstract: Let $ {B^n} = \{ z \in {{\mathbf{C}}^n}:\vert z\vert < 1\} $, $ {S^n} = \partial {B^n}$ and let $ {\sigma _n}$ be the Haar measure on $ {S^n}$. Then for all $ f \in {H^p}(1 \leqslant p < \infty )$ such that $ \operatorname{Im} (f(0)) = 0$ and $ t > 0$,

$\displaystyle {\sigma _n}(\{ z \in {S^n}:\vert f(z)\vert \geqslant t\} ) \leqslant {C_p} \cdot \frac{{\vert\vert\operatorname{Re} \,f\vert\vert _p^p}} {{{t^p}}}$

for some constant $ {C_p}$ depending only on $ p$. The best constant $ {C_p}$ is found for $ 1 \leqslant p \leqslant 2$.

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

  • [1] A. B. Aleksandrov, The existence of inner functions in the ball, Mat. Sb. (N.S.) 118 (1982), 147-163 (Russian); Math. USSR-Sb. 43 (1983), 143-159. MR 658785 (83i:32002)
  • [2] A. Baernstein, Some sharp inequalities for conjugate functions, Indiana Univ. Math. J. 27 (1978), 833-852. MR 503717 (80g:30022)
  • [3] B. Davis, On the weak-type $ (1 - 1)$ inequality for conjugate functions, Proc. Amer. Math. Soc. 44 (1974), 307-311. MR 0348381 (50:879)
  • [4] A. N. Kolmogorov, Sur les fonctions harmoniques conjugées et la séries de Fourier, Fund. Math. 7 (1925), 23-28.
  • [5] W. Rudin, Function theory in the unit ball of $ {{\mathbf{C}}^n}$, Springer-Verlag, New York, 1980. MR 601594 (82i:32002)
  • [6] B. Tomaszewski, The best constant in a weak-type $ {H^1}$-inequality, complex analysis and its applications, Complex Variables, Vol. 4, 1984, pp. 35-38. MR 770984 (86e:30003)
  • [7] A. Zygmund, Trigonometric series, Cambridge Univ. Press, London and New York, 1968. MR 0236587 (38:4882)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0801337-6
Article copyright: © Copyright 1985 American Mathematical Society

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