## Fejér means of Vilenkin-Fourier series.(English)Zbl 1265.42099

Let $$\{m_k\}_{k=0}^\infty$$ be a sequence of natural numbers that are greater than $$1$$, let $$M_0=1$$, $$M_{k+1}=M_km_k$$, $$k\geq 1$$, and let $$\mathbb Z_{m_k}$$ be the $$m_k$$-th cyclic group with discrete topology and measure $$\mu(x)=1/m_k$$ for all $$x\in \mathbb Z_{m_k}$$. Then $$G_m:=\prod^\infty_{k=0}\mathbb Z_{m_k}$$ is a compact Vilenkin group. The elements of $$G_m$$ are represented by sequences $$x=(x_1,x_2,\dots)$$, $$x_j\in\mathbb Z_{m_j}$$, and $$I_n(x)=\{y\in G_m: y_0=x_0,\dots,y_{n-1}=x_{n-1}\}$$. The $$\sigma$$-algebra generated by $$\{I_n(x):x\in G_m\}$$ is denoted by $$F_n$$. Let $$f=\{f^{(n)}\}_{n=0}^\infty$$ be a martingale with respect to $$\{F_n\}^\infty_{n=0}$$. Then $$f^*=\sup_{n\geq 0}| f^{(n)}|$$ is the maximal function of $$f$$. The Hardy martingale space $$H^p(G_m)$$ consists of all martingales $$f$$ such that $$\| f\| _{H^p}=\| f^*\| _{L^p(G_m)}<\infty$$, $$0<p<\infty$$. If $$\{\psi_i\}^\infty_{i=0}$$ is the character system for $$G_m$$, then $$\hat{f}(i):=\lim_{k\to\infty}\int_{G_m}f^{(k)}(x)\overline{\psi_i(x)}\,d\mu(x)$$, where $$\mu$$ is the normalized Haar measure on $$G_m$$, and $$\sigma_n(f)=\sum\limits^{n-1}_{i=0}(1-i/n)\hat{f}(i)\chi_i$$, $$n\geq 1$$, $$\sigma^*f(x)=\sup\limits_{n\geq 1}| \sigma_n(f)(x)|$$. The main result of the paper is the following.
There exists $$f\in H^{1/2}(G_m)$$ such that $$\sup\limits_{n\geq 1}\| \sigma_n(f)\| _{1/2}=\infty$$ and $$\| \sigma^*(f)\| _{1/2}=\infty$$, where $$\| \cdot\| _{1/2}$$ is the quasinorm in $$L^{1/2}(G_m)$$.

### MSC:

 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 60G42 Martingales with discrete parameter 60G46 Martingales and classical analysis
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