Maximal operators of Fejér means of double Vilenkin–Fourier series

Abstract

The main aim of this paper is to prove that the maximal operator σ∗ 0 := sup n |σn,n| of the Fejer means of the double Vilenkin-Fourier series is not bounded from the Hardy space H1/2 to the space weak-L1/2. Let N+ denote the set of positive integers, N := N+ ∪ {0}. Let m := (m0,m1, ...) denote a sequence of positive integers not less than 2. Denote by Zmk := {0, 1, ...,mk−1} the additive group of integers modulo mk. De ne the group Gm as the complete direct product of the groups Zmj , with the product of the discrete topologies of Zmj 's. The direct product μ of the measures μk({j}) := 1 mk (j ∈ Zmk) is the Haar measure on Gm with μ(Gm) = 1. If the sequencem is bounded, then Gm is called a bounded Vilenkin group, else its name is an unbounded one. The elements of Gm can be represented by sequences x := (x0, x1, ..., xj, ...) (xj ∈ Zmj). It is easy to give a base for the neighborhoods of Gm : I0(x) := Gm, In(x) := {y ∈ Gm|y0 = x0, ..., yn−1 = xn−1} for x ∈ Gm, n ∈ N. De ne In := In(0) for n ∈ N+. If we de ne the so-called generalized number system based on m in the following way: M0 := 1,Mk+1 := mkMk(k ∈ N), then every n ∈ N can be uniquely expressed as n = ∞ ∑ j=0 njMj, where nj ∈ Zmj (j ∈ N+) and only a nite number of nj's di er from zero. We use the following notations. Let (for n > 0) |n| := max{k ∈ N : nk 6= 0} (that is,M|n| ≤ n < M|n|+1), n = ∑∞ j=k njMj and n(k) := n− n(k). Denote by L(Gm) the usual (one dimensional) Lebesgue spaces (‖ · ‖p the corresponding norms) (1 ≤ p ≤ ∞). Next, we introduce on Gm an orthonormal system which is called the Vilenkin system. At rst de ne the complex valued functions rk(x) : Gm → C, the generalized Rademacher 2000 Mathematics Subject Classi cation. 42C10.