Rocky Mountain Journal of Mathematics

Derivatives of Blaschke products in weighted mixed norm spaces

Atte Reijonen

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For $1/2\lt p\lt \infty $, $0\lt q\lt \infty $ and a certain two-sided doubling weight $\omega $, we give a condition for the zeros of a Blaschke product $B$ which guarantees that $$ \|B'\|_{A^{p,q}_\omega }^q=\int _0^1 \bigg (\int _0^{2\pi } |B'(re^{i\theta })|^p d\theta \bigg )^{q/p} \omega (r)\,dr\lt \infty . $$ In addition, it is shown that the condition is necessary if the zero-sequence is a finite union of separated sequences.


This research was supported in part by Academy of Finland project Nos. 268009 and 286877, The Finnish Cultural Foundation and a JSPS Postdoctoral Fellowship for North American and European Researchers.

Article information

Rocky Mountain J. Math., Volume 49, Number 2 (2019), 627-643.

Received: 22 October 2017
First available in Project Euclid: 23 June 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30J10: Blaschke products
Secondary: 30J05: Inner functions

Blaschke product doubling weight inner function mixed norm space separated sequence


Reijonen, Atte. Derivatives of Blaschke products in weighted mixed norm spaces. Rocky Mountain J. Math. 49 (2019), no. 2, 627--643. doi:10.1216/RMJ-2019-49-2-627.

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  • P. Ahern, The mean modulus and the derivative of an inner function, Indiana Univ. Math. J. 28 (1979), 311–347.
  • ––––, The Poisson integral of a singular measure, Canadian J. Math. 35 (1983), 735–749.
  • P.R. Ahern and D.N. Clark, On inner functions with $B^p$ derivative, Michigan Math. J. 23 (1976), 107–118.
  • A. Aleman and D. Vukotić, On Blaschke products with derivatives in Bergman spaces with normal weights, J. Math. Anal. Appl. 361 (2010), 492–505.
  • B. Böe, A norm on the holomorphic Besov space, Proc. Amer. Math. Soc. 131 (2003), 235–241.
  • W.S. Cohn, Radial limits and star invariant subspaces of bounded mean oscillation, Amer. J. Math. 108 (1986), 719–749.
  • P. Colwell, Blaschke products: Bounded analytic functions, University of Michigan Press, Ann Arbor, Michigan, 1985.
  • P. Duren, Theory of $H^p$ spaces, Academic Press, New York, 1970.
  • D. Girela, C. González and M. Jevtić, Inner functions in Lipschitz, Besov, and Sobolev spaces, Abstract Appl. Anal. 2011, article ID 626254.
  • A. Gluchoff, On inner functions with derivative in Bergman spaces, Illinois J. Math. 31 (1987), 518–528.
  • J. Gröhn and A. Nicolau, Inner functions in certain Hardy-Sobolev spaces, J. Funct. Anal. 272 (2017), 2463–2486.
  • H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces, Grad. Texts Math. 199 (2000).
  • M. Jevtić, On Blaschke products in Besov spaces, J. Math. Anal. Appl. 149 (1990), 86–95.
  • H.O. Kim, Derivatives of Blaschke products, Pacific J. Math. 114 (1984), 175–190.
  • J. Mashreghi, Derivatives of inner functions, Fields Inst. Mono. 31 (2013).
  • J.A. Peláez and J. Rättyä, Embedding theorems for Bergman spaces via harmonic analysis, Math. Ann. 362 (2015), 205–239.
  • ––––, Bergman projection induced by radial weight, preprint.
  • F. Pérez-González and J. Rättyä, Derivatives of inner functions in weighted Bergman spaces and the Schwarz-Pick lemma, Proc. Amer. Math. Soc. 145 (2017), 2155–2166.
  • F. Pérez-González, J. Rättyä and A. Reijonen, Derivatives of inner functions in Bergman spaces induced by doubling weights, Ann. Acad. Sci. Fenn. Math. 42 (2017), 735–753.
  • A. Reijonen, Derivatives of Blaschke products whose zeros lie in a Stolz domain and weighted Bergman spaces, Proc. Amer. Math. Soc. 146 (2018), 1173–1180.
  • ––––, Derivatives of inner functions in weighted mixed norm spaces, J. Geom. Anal..
  • I.È. Verbitski\v\i, Inner functions, Besov spaces and multipliers, Dokl. Akad. Nauk 276 (1984), 11–14.