Generalized Hardy inequalities and pseudocontinuable functions
Let \(H^1\) be the Hardy space on the unit circle \({\mathbf T}\), i.e. subspace of \(L^1({\mathbf T})\) consisting of all functions \(f\) with the spectrum \(\text{spec }f=\{n\in\mathbb{Z}:\widehat f(n)\neq 0\}\) in \([0,\infty)\), where \(\widehat f\) denotes the Fourier transform of \(f\). Let \((n_k)^\infty_1\) be an increasing sequence of positive integers. The generalized Hardy inequality states, that for some (absolute) constant \(C>0\), one has \[ \sum^\infty_{k=1} {|\widehat f(n_k)|\over k}\leq C|f|_1, \] whenever \(f\) is an \(H^1\) function satisfying the indispensable condition \(\text{spec }f\subset(n_k)^\infty_1\). The author refuses the latter condition and considers some cases where the above-mentioned inequality holds true. For \(p\), \(1\leq p<\infty\), set \[ \Lambda((n_k),p)= \sup\Biggl\{\sum^\infty_{k=1}|\widehat f(n_k)|^pk^{-1}:f\in H^1, |f|_1=1\Biggr\}. \] It is proved: (i) If \(inf_k{k\over n_k}(n_{k+1}-n_k)= \delta>0\), then \(\Lambda((n_k,p)\leq \text{const}(1+\delta^{-1})\), where the constant is numerical; (ii) given any decreasing sequence of positive numbers \((\delta_k)^\infty_1\) with \(\lim_k\delta_k= 0\), there exist positive integers \(n_1<n_2<\cdots\) such that \({k\over n_k}(n_{k+1}- n_k)\geq \delta_k\), \(k=1,2,\dots\), and \(\Lambda((n_k),p)=\infty\) for all \(p\), \(1\leq p<\infty\). In addition, it is given an example of the infinite-dimensional subspace of \(H^1\) for whose elements the generalized Hardy inequality is always valid, no additional restrictions on the \(\text{spec }f\) and \(n_k\)'s being imposed. The rest of the paper is related to the pseudocontinuable functions, where a result of \textit{R. G. Douglas}, \textit{H. S. Shapiro} and \textit{A. L. Shields} [Ann. Inst. Fourier 20, No. 1, 37-76 (1970; Zbl 0186.45302)] on the spectrum of a pseudocontinuable function is extended.
scientific article; zbMATH DE number 944331