Abstract:

In this paper, we discuss the boundedness of maximal operator with respect to bounded Vilenkin-like system (or $\psi\alpha$ system) which is generalization of bounded Vilenkin system. We prove that when $0 < p <1/2$ the maximal operator $\tilde{\sigma}_p^*f=\sup\limits_{n\in {\Bbb N}}\frac{|\sigma_nf|}{(n+1)^{1/p-2}}$ is bounded from the martingale Hardy space $H_p$ to the space $L_p$, where $\sigma_nf$ is $n$-th Fej\'er mean with respect to bounded Vilenkin-like system. By a counterexample, we also prove that the maximal operator $\sup\limits_{n\in {\Bbb N}}\frac{|\sigma_nf|}{\varphi(n)}$ is not bounded from the martingale Hardy space $H_{p}$ to the space $L_{p,\infty}$ when $0 < p <1/2$ and $\mathop{\overline{\lim}}\limits_{n\rightarrow \infty}\frac{(n+1)^{1/p-2}}{\varphi(n)}=+\infty$.

Key words: Vilenkin-like system, Maximal operator, Fejér mean, Hardy spaces