Sharp \((H_p, L_p)\) type inequalities of maximal operators of \(T\) means with respect to Vilenkin systems (Q2117370)

Let \(f=(f^{(n)})_{n\in \mathbb{N}}\) be a martingale on a bounded type Vilenkin group and \(S_k f\) \((k=0,1,\dots)\) be the partial sums of the Vilenkin-Fourier series of \(f\). For a sequence of non-negative numbers \((q_k)\) with \(q_0>0\) the means \(T_n\) of the Vilenkin-Fourier series of \(f\) are defined as follows \[ T_n f=\frac{1}{Q_n}\sum_{k=0}^{n-1} q_k S_k f, \] where \(Q_n=q_0 +q_1 + \dots + q_{n-1}\). Suppose \((q_k)\) is either a non-increasing sequence or a non-decreasing sequence satisfying the condition \(q_{n-1}/Q_n=O(1/n)\) \((n\rightarrow \infty)\). It is proved that for every \(p\in (0, 1/2]\) the following weighted maximal operator \[ M_pf=\sup_n\frac{|T_n f|}{(n+1)^{1/p -2} \log ^{2[1/2+p]} (n+1)} \] is bounded from the martingale Hardy space \(H_p\) to the space \(L_p\). Suppose \((q_k)\) is a monotone sequence. For the parameters \(p\in(0,1/2)\) it is obtained the following estimate for \(L_p\)-norms of the means \(T_k f\) \[ \sum_{k=1}^{\infty}\frac{||T_k f||_p^{p}}{k^{2-2p}}\leq C_p||f||_{H_p}^{p}. \] For the endpoint case \(p=1/2\) under some additional restrictions on the sequence \((q_k)\) it is shown the estimate \[ \sum_{k=1}^{n}\frac{||T_k f||_{1/2}^{1/2}}{k \log n}\leq C||f||_{H_{1/2}}^{1/2} \;\;\;\;(n\in \mathbb{N}). \]