Abstract
We prove and discuss some new $H_{p}$ - $L_{p}$ type inequalities of weighted maximal operators of Vilenkin-Norlund means with monotone coefficients. It is also proved that these inequalities are the best possible in a special sense. We also apply these results to prove strong summability for such Vilenkin-Norlund means. As applications, both some well-known and new results are pointed out.
Highlights
The definitions and notations used in this introduction can be found in our section
In the one-dimensional case the weak (, )-type inequality for maximal operator of Fejér means σ ∗f := supn∈N |σnf | can be found in Schipp [ ] for Walsh series and in Pál, Simon [ ] for bounded Vilenkin series
In the case p = / a counterexample with respect to Walsh system was given by Goginava [ ] and for the bounded Vilenkin system was proved by Tephnadze [ ]
Summary
The definitions and notations used in this introduction can be found in our section. Weisz [ ] proved that the maximal operator of the Fejér means σ ∗ is bounded from the Hardy space H / to the space weak-L /. Weisz [ ] proved that the maximal operator of Cesàro means σ α,∗f := supn∈N |σnαf | is bounded from the martingale space Hp to the space Lp for p > /( +α). Simon and Weisz [ ] showed that the maximal operator σ α,∗ ( < α < ) of the (C, α) means is bounded from the Hardy space H /( +α) to the space weak-L /( +α). Simon [ ] (for unbounded Vilenkin systems in the case when p = see [ ] and for < p < another proof was pointed out in [ ]) proved that there exists an absolute constant cp, depending only on p, such that n log[p] n k=.
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