Abstract: For Vilenkin-like system, the authors define a new operator H* f:=sup_n|H_nf|, where H_nf is the weighted average for partial sums, and prove that H* is of type (H*_p(G_m), L_p(G_m)) for all 1/2 < p ≤ ∞ . As a consequence, the authors prove the operator S*f:=sup_n|S_nf| is of type (p, p) for 1<p< ∞, where S_nf is the n-partial sum.

Key words: Martingale Hardy space, Vilenkin-like systems, weighted average