On multipliers in Hardy spaces (Q2754803)

Let \(M_q\) be the Banach space of multipliers in the Hardy space \(H_q\), \(0<q<2\). Let \(M_q[0,\infty)\) be the space of functions \(\varphi\) on \([0,\infty)\) with a finite norm \(\sup_{\varepsilon >0}\left\|\{\varphi (\varepsilon k)\}_{k=0}^\infty \right\|_{M_q}\). It is known that any function \(\varphi\) which vanishes on \([1,\infty)\) and has a bounded variation on \([0,1]\) belongs to \(M_q[0,\infty)\). The author shows that the boundedness of the variation cannot be replaced by any growth rate of the variation on \([x,1]\) as \(x\to +0\).