Hardy空间上维林肯型系统下的极大算子

The Maximal Operator of Vilenkin-like System on Hardy Spaces

维林肯型系统(或ψα系统)是维林肯系统的推广,该文研究有界维林肯型系统下的极大算子的有界性.该文证明当0<p<1/2时,极大算子σ^(∗)pf=supn∈N|σnf|(n+1)^(1)/p−2是从鞅Hardy空间Hp到Lp有界的,其中σnf是关于有界维林肯型系统的Fej\'er均值.并通过构造反例,证明当0<p<1/2且lim¯¯¯¯¯¯¯n→∞(n+1)^(1)/p−2φ(n)=+∞时,极大算子supn∈N|σnf|φ(n)不是从鞅Hardy空间Hp到L_(p,∞)有界的.

In this paper,we discuss the boundedness of maximal operator with respect to bounded Vilenkin-like system(orψαsystem)which is generalization of bounded Vilenkin system.We prove that when 0<p<1/2 the maximal operatorσ^(∗)pf=supn∈N|σnf|(n+1)^(1)/p−2 is bounded from the martingale Hardy space Hp to the space Lp,whereσnf is n-th Fej\'er mean with respect to bounded Vilenkin-like system.By a counterexample,we also prove that the maximal operator supn∈N|σnf|φ(n)is not bounded from the martingale Hardy space Hp to the space L_(p,∞)when 0<p<1/2 and lim¯¯¯¯¯¯¯n→∞(n+1)^(1)/p−2φ(n)=+∞.