Hardy-Littlewood type inequalities for Vilenkin-Fourier coefficients
The Hardy type inequality \[ \left(\sum_{k=3D1}^{\infty}\frac{| \widehat f(k)| ^p}{k^{2-p}}\right) \leq C_p \| f\| _{H^p_{**}} \quad (1/2 < p\leq 2) \tag{\(\ast\)} \] is proved for functions \(f\) belonging to the Hardy space \(H^p_{**}(G_m)\) defined by means of a maximal function. The authors extend (\(\ast\)) for \(2<p<\infty\) when the Vilenkin-Fourier coefficients of \(f\) are \(\lambda\)-blockwise monotone. They show that under some conditions on the Vilenkin system (in particular, for some unbounded type, too) a converse version of (\(\ast\)) holds also for all \(p>0\) provided that the Vilenkin-Fourier coefficients of \(f\) are monotone.
scientific article; zbMATH DE number 1261156
scientific article