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Derivatives of Hardy functions


Author: Boo Rim Choe
Journal: Proc. Amer. Math. Soc. 110 (1990), 781-787
MSC: Primary 32A35
DOI: https://doi.org/10.1090/S0002-9939-1990-1028041-8
MathSciNet review: 1028041
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Abstract: Let $ B$ be the open unit ball of $ {C^n}$, and set $ S = \partial B$. It is shown that if $ \varphi \in {L^p}\left( S \right),\varphi > 0$, is a lower semicontinuous function on $ S$ and $ 1/q > 1 + 1/p$, then, for a given $ \varepsilon > 0$, there exists a function $ f \in {H^p}\left( B \right)$ with $ f\left( 0 \right) = 0$ such that $ \left\vert {{f^ * }} \right\vert = \varphi $ almost everywhere on $ S$ and $ \int_B {{{\left\vert {\nabla f} \right\vert}^q}dV < \varepsilon } $ where $ V$ denotes the normalized volume measure on $ B$.


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

  • [1] A. B. Alexandrov, Existence of inner functions in the unit ball, Mat. Sb. 118 (1982), 147-163. MR 658785 (83i:32002)
  • [2] E. Bedford and A. B. Taylor, Two applications of a nonlinear integral formula to analytic functions, Indiana Univ. Math. J. 29 (1980), 463-465. MR 570694 (81e:32020)
  • [3] B. R. Choe, Projections, the weighted Bergman spaces, and the Bloch space, Proc. Amer. Math. Soc. 108 (1990), 127-136. MR 991692 (90h:32009)
  • [4] W. Rudin, Function theory in the unit ball of $ {{\text{C}}^n}$, Springer, New York, 1980. MR 601594 (82i:32002)
  • [5] -, Inner functions in the unit ball of $ {{\text{C}}^n}$, J. Funct. Anal. 50 (1983), 100-126. MR 690001 (84i:32007)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1028041-8
Keywords: Derivatives, $ {H^p}$-functions
Article copyright: © Copyright 1990 American Mathematical Society

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