In the article under review, the authors derive the maximal subspace of natural numbers \(\left\{n_k: k \geq 0\right\}\), such that the restricted maximal operator, defined by \(\sup _{k \in \mathbb{N}}\left|\sigma_{n_k} F\right|\) on this subspace of Fejér means of Walsh-Fourier series is bounded from the martingale Hardy space \(H_{1 / 2}\) to the Lebesgue space \(L_{1 / 2}\). The sharpness of this result is also proved in the article.
Let \(\mathbb{N}_{+}\) denote the set of the positive integers, \(\mathbb{N}:=\mathbb{N}_{+} \cup\{0\}\). Denote by \(Z_2\) the discrete cyclic group of order 2, that is \(Z_2:=\{0,1\}\), where the group operation is the modulo 2 addition and every subset is open. The Haar measure on \(Z_2\) is given so that the measure of a singleton is \(1 / 2\).
Define the group \(G\) as the complete direct product of the group \(Z_2\), with the product of the discrete topologies of \(Z_2\). The elements of \(G\) are represented by sequences \(x:=\left(x_0, x_1, \ldots, x_j, \ldots\right)\), where \(x_k=0 \vee 1\).
If \(n \in \mathbb{N}\), then every \(n\) can be uniquely expressed as \(n=\sum_{j=0}^{\infty} n_j 2^j\), where \(n_j \in Z_2 \quad(j \in \mathbb{N})\) and only a finite numbers of \(n_j\) differ from zero. Every \(n \in \mathbb{N}\) can be also represented as \[ n=\sum_{i=1}^r 2^{n_i}, n_1>n_2>\cdots n_r \geq 0. \] For such a representation of \(n \in \mathbb{N}\), we denote numbers \[ n^{(i)}=2^{n_{i+1}}+\cdots+2^{n_r}, \quad i=1, \ldots, r. \] Let \(2^s \leq n_{s_1} \leq n_{s_2} \leq \cdots \leq n_{s_r} \leq 2^{s+1}, s \in \mathbb{N}\). For such \(n_{s_j}\), which can be written as \[ n_{s_j}=\sum_{i=1}^{r_{s_j}} \sum_{k=l_i^{s_j}}^{t_i^{s_j}} 2^k, \] where \(0 \leq l_1^{s_j} \leq t_1^{s_j} \leq l_2^{s_j}-2<l_2^{s_j} \leq t_2^{s_j} \leq \cdots \leq l_{r_j}^{s_j}-2<l_{r_{s_j}}^{s_j} \leq t_{r_{s_j}}^{s_j}\), we define \begin{align*} A_s: & =\bigcup_{j=1}^r\left\{l_1^{s_j}, t_1^{s_j}, l_2^{s_j}, t_2^{s_j}, \ldots, l_{r_{s_j}}^{s_j}, t_{r_{s_j}}^{s_j}\right\}\\ & =\left\{l_1^s, l_2^s, \ldots, l_{r_s^1}^s\right\} \bigcup\left\{t_1^s, t_2^s, \ldots, t_{r_s^2}^s\right\}=\left\{u_1^s, u_2^s, \ldots, u_{r_s^3}^s\right\}\tag{cond2} \end{align*} where \(u_1^s<u_2^s<\cdots<u_{r_s^3}^s\). We note that \(t_{r_{s_j}}^{s_j}=s \in A_s\), for \(j=1,2, \ldots, r\). We denote the cardinality of the set \(A_s\) by \(\left|A_s\right|\), that is \[ \operatorname{card}\left(A_s\right):=\left|A_s\right| . \] The \(k\)-th Rademacher function is defined by \[ r_k(x):=(-1)^{x_k} \quad(x \in G, k \in \mathbb{N}). \] Then, the Walsh system \(w:=\left(w_n: n \in \mathbb{N}\right)\) on \(G\) is defined as \[ w_n(x):=\prod_{k=0}^{\infty} r_k^{n_k}(x)=r_{|n|}(x)(-1)^{\sum_{k=0}^{|n|-1} n_k x_k} \quad(n \in \mathbb{N}) . \] The Walsh system is orthonormal and complete in \(L_2(G)\) (see [F. Schipp et al., Walsh series: an introduction to dyadic harmonic analysis. Bristol, England: Adam Hilger (1990)]). If \(f \in L_1(G)\), then we can define the Fourier coefficients, \[ \widehat{f}(n) :=\int_G f w_n \mathrm{~d} \mu, \quad(n \in \mathbb{N}), \] the partial sums of Fourier series, \[ S_n f :=\sum_{k=0}^{n-1} \widehat{f}(k) w_k, \quad\left(n \in \mathbb{N}_{+}, S_0 f:=0\right), \] and the Fejér means, \[ \sigma_n f :=\frac{1}{n} \sum_{k=1}^n S_k f. \] The maximal operator \(\sigma^\ast\) of Fejér means \(\sigma_n\) with respect to the Walsh system, is defined \[ \sigma^\ast f:=\sup _{n \in \mathbb{N}}\left|\sigma_n f\right| \] F. Sipp [Mat. Zametki 18, 193–201 (1975; Zbl 0349.42013)] and J. Pal and P. Simon [Acta Math. Acad. Sci. Hung. 29, 155–164 (1977; Zbl 0345.42011)] studied the weak \((1,1)\)-type inequality for \(\sigma^* \). N. Fujii [Proc. Am. Math. Soc. 77, 111–116 (1979; Zbl 0415.43014)] and P. Simon [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 27, 87–101 (1984; Zbl 0586.43001)] proved that \(\sigma^\ast\) is bounded from \(H_1\) to \(L_1\). F. Weisz [Anal. Math. 22, No. 3, 229–242 (1996; Zbl 0866.42020)] generalized this result and proved the boundedness of \(\sigma^*\) from the martingale space \(H_p\) to the Lebesgue space \(L_p\) for \(p>1 / 2\). P. Simon [Monatsh. Math. 131, No. 4, 321–334 (2001; Zbl 0976.42014)] gave a counterexample, which shows that boundedness does not hold for \(0<p<1 / 2\). A counterexample for \(p=1 / 2\) was given by U. Goginava [Acta Math. Hung. 115, No. 4, 333–340 (2007; Zbl 1174.42336)]. Moreover, in [Acta Math. Sin., Engl. Ser. 27, No. 10, 1949–1958 (2011; Zbl 1282.42026)], he proved that there exists a martingale \(F \in H_p, \; 0<p \leq 1 / 2\), such that \[ \sup _{n \in \mathbb{N}}\left\|\sigma_n F\right\|_p=+\infty . \] F. Weisz [Anal. Math. 30, No. 2, 147–160 (2004; Zbl 1060.42022)] proved that the maximal operator \(\sigma^\ast\) of the Fejér means is bounded from the Hardy space \(H_{1 / 2}\) to the space weak- \(L_{1 / 2}\).
To study the convergence of subsequences of Fejér means and their restricted maximal operators on the martingale Hardy spaces \(H_p(G)\) for \(0<p \leq 1 / 2\), the central role is played by the fact that any natural number \(n \in \mathbb{N}\) can be uniquely expression as \(n=\sum_{k=0}^{\infty} n_j 2^j\), \(\quad n_j \in Z_2\) \((j \in \mathbb{N})\), where only a finite numbers of \(n_j\) differ from zero and their important characters \([n],|n|, \rho(n)\) and \(V(n)\) are defined by \[ [n]:=\min \left\{j \in \mathbb{N}, n_j \neq 0\right\}, \quad|n|:=\max \left\{j \in \mathbb{N}, n_j \neq 0\right\}, \quad \rho(n)=|n|-[n] \] and \[ V(n):=n_0+\sum_{k=1}^{\infty}\left|n_k-n_{k-1}\right|, \quad \text { for all } n \in \mathbb{N}. \] F. Weisz [Summability of multi-dimensional Fourier series and Hardy spaces. Dordrecht: Springer (2002; Zbl 1306.42003)] (see also [F. Weisz, Martingale Hardy spaces and their applications in Fourier analysis. Berlin: Springer-Verlag (1994; Zbl 0796.60049)]) also proved that for any \(F \in H_p(G)\) \((p>0)\), the maximal operator \(\sup _{n \in \mathbb{N}}\left|\sigma_{2^n} F\right|\) is bounded from the Hardy space \(H_p\) to the Lebesgue space \(L_p\). Furthermore, in [L. E. Persson and G. Tephnadze, Mediterr. J. Math. 13, No. 4, 1841–1853 (2016; Zbl 1358.42023)] this result was generalized and it was proved that if \(0<p \leq 1 / 2\) and \(\left\{n_k: k \geq 0\right\}\) is a sequence of positive numbers, such that \[ \sup _{k \in \mathbb{N}} \rho\left(n_k\right) \leq c<\infty\tag{cond} \] then the maximal operator \(\widetilde{\sigma}^{\ast, \nabla}\), defined by \[ \tilde{\sigma}^{\ast, \nabla} F=\sup _{k \in \mathbb{N}}\left|\sigma_{n_k} F\right| \] is bounded from the Hardy space \(H_p\) to the Lebesgue space \(L_p\). Moreover, if \(0<p<\) \(1 / 2\) and \(\left\{n_k: k \geq 0\right\}\) is a sequence of positive numbers, such that \(\sup _{k \in \mathbb{N}} \rho\left(n_k\right)=\infty\), then there exists a martingale \(F \in H_p\) such that \[ \sup _{k \in \mathbb{N}}\left\|\sigma_{n_k} F\right\|_p=\infty. \]
From this fact, it follows that if \(0<p<1 / 2\), \(f \in H_p\) and \(\left\{n_k: k \geq 0\right\}\) is any sequence of positive numbers, then \(\sigma_{n_k} f\) are uniformly bounded from the Hardy space \(H_p\) to the Lebesgue space \(L_p\) if and only if the condition (cond) is fulfilled. Moreover, condition (cond) is a necessary and sufficient condition for the boundedness of subsequence \(\sigma_{n_k} f\) from the Hardy space \(H_p\) to the Hardy space \(H_p\).
G. Tephnadze [J. Contemp. Math. Anal., Armen. Acad. Sci. 51, No. 2, 90–102 (2016; Zbl 1358.42024)] proved some results which in particular, imply that if \(f \in H_{1 / 2}\) and \(\left\{n_k: k \geq 0\right\}\) is any sequence of positive numbers, then \(\sigma_{n_k} f\) are bounded from the Hardy space \(H_{1 / 2}\) to the space \(H_{1 / 2}\) if and only if, for some \(c\), \[ \sup _{k \in \mathbb{N}} V\left(n_k\right)<c<\infty. \] In the article under review, the authors investigate the limit case \(p=1 / 2\). In particular, they derive the maximal subspace of natural numbers \(\left\{n_k: k \geq 0\right\}\), such that restricted maximal operator, defined by \(\sup _{k \in \mathbb{N}}\left|\sigma_{n_k} F\right|\) on this subspace of Fejér means of Walsh-Fourier series is bounded from the martingale Hardy space \(H_{1 / 2}\) to the Lebesgue space \(L_{1 / 2}\).
The main result obtained is the following: Theorem.

(a)

Let \(f \in H_{1 / 2}(G)\) and \(\left\{n_k: k \geq 0\right\}\) be a sequence of positive numbers and let \(\left\{n_{s_i}: 1 \leq i \leq r\right\} \subset\left\{n_k: k \geq 0\right\}\) be numbers such that \(2^s \leq n_{s_1} \leq\) \(n_{s_2} \leq \cdots \leq n_{s_r} \leq 2^{s+1}, s \in \mathbb{N}\). If the sets \(A_s\), defined by (cond2), are uniformly finite for all \(s \in \mathbb{N}\), that is the cardinality of the sets \(A_s\) are uniformly finite: \[ \sup _{s \in \mathbb{N}}\left|A_s\right|<c<\infty, \] then the restricted maximal operator \(\tilde{\sigma}^{\ast, \nabla}\), defined by \[ \widetilde{\sigma}^{\ast, \nabla} F=\sup _{k \in \mathbb{N}}\left|\sigma_{n_k} F\right| \] is bounded from the Hardy space \(H_{1 / 2}\) to the Lebesgue space \(L_{1 / 2}\).

(b)

(Sharpness) Let \[ \sup _{s \in \mathbb{N}}\left|A_s\right|=\infty. \] Then, there exists a martingale \(f \in H_{1 / 2}(G)\), such that the maximal operator, defined by (3.1), is not bounded from the Hardy space \(H_{1 / 2}\) to the Lebesgue space \(L_{1 / 2}\).

In particular, Theorem 1 implies the following optimal characterization:
Corollary. Let \(F \in H_{1 / 2}(G)\) and \(\left\{n_k: k \geq 0\right\}\) be a sequence of positive numbers. Then, the restricted maximal operator \(\tilde{\sigma}^{\ast, \nabla}\), defined by (3.1), is bounded from the Hardy space \(H_{1 / 2}\) to the Lebesgue space \(L_{1 / 2}\) if and only if any sequence of positive numbers \(\left\{n_k: k \geq 0\right\}\) which satisfies \(n_k \in\left[2^s, 2^{s+1}\right)\), is uniformly finite for each \(s \in \mathbb{N}_{+}\) and each \(\left\{n_k: k \geq 0\right\}\) has bounded variation, i.e., \[ \sup _{k \in \mathbb{N}} V\left(n_k\right)<c<\infty. \] In order to be able to compare with some other results in the literature they also state the following:
Corollary. Let \(F \in H_{1 / 2}(G)\). Then, the restricted maximal operators \(\widetilde{\sigma}_i^{\ast, \nabla}, i=\) \(1,2,3\), defined by \[ \begin{aligned} & \tilde{\sigma}_1^{\ast, \nabla} F=\sup _{k \in \mathbb{N}}\left|\sigma_{2^k} F\right|, \\ & \widetilde{\sigma}_2^{\ast, \nabla} F=\sup _{k \in \mathbb{N}}\left|\sigma_{2^k+1} F\right|, \\ & \tilde{\sigma}_3^{\ast, \nabla} F=\sup _{k \in \mathbb{N}}\left|\sigma_{2^k+2^{[k / 2]}} F\right|, \end{aligned} \] where \([n]\) denotes the integer part of \(n\), are all bounded from the Hardy space \(H_{1 / 2}\) to the Lebesgue space \(L_{1 / 2}\).
In [L. E. Persson and G. Tephnadze, Mediterr. J. Math. 13, No. 4, 1841–1853 (2016; Zbl 1358.42023)], it was proved that if \(0<p<1 / 2\), then the restricted maximal operators \(\widetilde{\sigma}_2^{\ast, \nabla}\) and \(\widetilde{\sigma}_3^{\ast, \nabla}\), are not bounded from the Hardy space \(H_p\) to the Lebesgue space weak \(-L_p\).
On the other hand, F. Weisz [Martingale Hardy spaces and their applications in Fourier analysis. Berlin: Springer-Verlag (1994; Zbl 0796.60049)] (see also [L. E. Persson and G. Tephnadze, Mediterr. J. Math. 13, No. 4, 1841–1853 (2016; Zbl 1358.42023)]) proved that if \(0<p \leq 1 / 2\), then the restricted maximal operator \(\widetilde{\sigma}_1^{\ast, \nabla}\), is bounded from the Hardy space \(H_p\) to the Lebesgue space \(L_p\).