Generalized Hardy inequalities and pseudocontinuable functions

Abstract

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 f∈H¹, provided that the n_k'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.