Arkiv för Matematik

  • Ark. Mat.
  • Volume 34, Number 2 (1996), 231-244.

Generalized Hardy inequalities and pseudocontinuable functions

Konstatin M. Dyakonov

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Given positive integers n1< n2<... we show that the Hardy-type inequality $\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f(n_k )} \right|}}{k}} \leqslant const\left\| f \right\|1$ holds true for all fH1, provided that the nk's, satisfy an appropriate (and indispensable) regularity condition. On the other hand, we exhibit inifinite-dimensional subspaces of H1 for whose elements the above inequality is always valid, no additional hypotheses being imposed. In conclusion, we extend a result of Douglas, Shapiro and Shields on the cyclicity of lacunary series for the backward shift operator.


Supported in part by Grants R2D000 and R2D300 from the International Science Foundation and by a grant from Pro Mathematica (France).

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Ark. Mat. Volume 34, Number 2 (1996), 231-244.

Received: 10 April 1995
First available in Project Euclid: 31 January 2017

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1996 © Institut Mittag-Leffler


Dyakonov, Konstatin M. Generalized Hardy inequalities and pseudocontinuable functions. Ark. Mat. 34 (1996), no. 2, 231--244. doi:10.1007/BF02559546.

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