Hardy-Littlewood inequalities for two-parameter Vilenkin-Fourier coefficients (Q2718026)

The authors continue to apply their techniques developed in [Anal. Math. 24, No. 2, 131-150 (1998; Zbl 0913.42020)]. They prove Vilenkin analogues of a classical inequality of Hardy and Littlewood, namely they prove that under certain conditions the double Vilenkin-Fourier coefficients of \(f\) satisfy NEWLINE\[NEWLINE\Biggl(\sum_{k, \ell\in\Gamma_\alpha} {|\widehat f(k,\ell)|^p\over (k\ell)^{2- p}}\Biggr)^{1/p}\leq C_p\|f\|_{H^p},NEWLINE\]NEWLINE where \(0< \alpha\leq 1\) and \(\Gamma_\alpha\) is a cone of aperture \(\alpha\), i.e., the set of \((x,y)\in [0,1]\times [0,1]\) such that \(\alpha\leq x/y\leq 1/\alpha\). The precise hypotheses depend, of course, on which Hardy space is used and how rapidly the ``generators'' \({\mathbf m}:=\{m_1, m_2,\dots\}\) of the Vilenkin system grow. For example, if \({\mathbf m}\) is bounded (or even quasi-monotone) or if \(H^p\) is the Hardy space generated by means of the conditional quadratic variation, then the estimate above holds for all \(0<\alpha\leq 1\). If, however, \({\mathbf m}\) is unbounded and \(H^p\) is the Hardy space defined by means of the diagonal maximal function, then one must insist that \(2/3\leq p\leq 1\). Examples are included to show that the critical index \(p= 2/3\) cannot be lowered. The dual inequalities for functions of bounded mean oscillation are also mentioned.