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Remark on integral means of derivatives of Blaschke products

  • Atte Reijonen,
    Department of Physics and Mathematics, University of Eastern Finland, P.O.Box 111, FI-80101 Joensuu, Finland
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Abstract

If $B$ is the Blachke product with zeros $\{z_n\}$, then $|B'(z)|\le \Psi_B(z)$, where $$\Psi_B(z)=\sum_n \frac{1-|z_n|^2}{|1-\overline{z}_nz|^2}.$$ Moreover, it is a well-known fact that, for $0\lt p\lt \infty$, $$M_p(r,B')= \left(\frac{1}{2\pi}\int_{0}^{2\pi} |B'(re^{i\t})|^p\,d\t \right)^{1/p}, \quad 0\le r\lt 1,$$ is bounded if and only if $M_p(r,\Psi_B)$ is bounded. We find a Blaschke product $B_0$ such that $M_p(r,B_0')$ and $M_p(r,\Psi_{B_0})$ are not comparable for any $\frac12\lt p\lt \infty$. In addition, it is shown that, if $0\lt p\lt \infty$, $B$ is a Carleson-Newman Blaschke product and a weight $\omega$ satisfies a certain regularity condition, then $$ \int_\mathbb{D} |B'(z)|^p\omega(z)\,dA(z)\asymp \int_\mathbb{D} \Psi_B(z)^p\omega(z)\,dA(z), $$ where $dA(z)$ is the Lebesgue area measure on the unit disc.
Keywords: Bergman space, Blaschke product, Hardy space, integral mean Bergman space, Blaschke product, Hardy space, integral mean
MSC Classifications: 30J10, 30H10, 30H20 show english descriptions Blaschke products
Hardy spaces
Bergman spaces, Fock spaces
30J10 - Blaschke products
30H10 - Hardy spaces
30H20 - Bergman spaces, Fock spaces
 

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