Weighted average of dirichlet kernels for vilenkin-like system

For Vilenkin-like system, the authors define a new operator H* f ≔ supn|Hnf|, where Hnf is the weighted average for partial sums, and prove that H* is of type (H*p(Gm), Lp(Gm)) for all ½ < p ≤ ∞. As a consequence, the authors prove the operator S* f ≔ |Snf| is of type (p,p) for 1 < p > ∞, where Snf is the n-partial sum.