Let \(G\) be the complete direct countable product of discrete groups \(\mathbb Z_2=\{0,1\}\) with product measure \(\mu\). Every \(x\in G\) has the form \(x=(x_0,x_1,\dots)\), \(x_i\in\mathbb Z_2\). If \(k\in {\boldsymbol P}=\{0,1,\dots\}\) and \(x\in G\), then \(r_k(x):=(-1)^{x_k}\). For \(n\in{\boldsymbol P}\) we have \(n=\sum^{| n| }_{i=0}n_i2^i\), where \(n_i\in\mathbb Z_2\), \(| n| =\max\{j\in{\boldsymbol P}:n_j\neq 0\}\). Then by definition \(\kappa_n(x)=r_{| n| }(x)\prod^{| n| -1}_{k=0}(r_{| n| -1-k} (x))^{n_k}\). System \(\{\kappa_n(x)\}_{n\in{\boldsymbol P}}\) is called Walsh-Kaczmarz system. For a dyadic martingale \(f=\{f^{(n)}\}_{n=1}^\infty\) one can define \[ \hat{f}_\kappa(n) =\lim_{n\to\infty}\int_G f^{(n)}(x) \kappa_n(x)\,d\mu(x),\;n\in{\boldsymbol P}, \quad S_n^\kappa(f,x)=\sum^{n-1}_{i=0}\hat{f}_\kappa(i)\kappa_i(x); \]
\[ \sigma_n^\kappa(f,x)=n^{-1}\sum_{k=1}^n S_k^\kappa(f,x), \quad n\in\mathbb N=\{1,2,\dots\}, \quad \sigma^{\kappa,*}_p f:=\sup_{n\in{\boldsymbol P}}(n+1)^{2-1/p}| \sigma_n^\kappa(f)| . \] If \(p>0\) and \(f^*:=\sup_{n\in{\boldsymbol P}}| f^{(n)}| \in L^p(G)\), then \(f\in H^p(G)\) and \(\| f\| _{H^p}=\| f^*\| _{L^p}\). Further \(\| f\| _{L^{p,\infty}} :=\sup_{\lambda>0} \lambda(\mu\{| f| >\lambda\})^{1/p}\).
Theorem 1. Let \(0<p<1/2\). Then the maximal operator \(\sigma^{\kappa,*}_p f\) is bounded from \(H^p(G)\) to \(L^p(G)\).
Theorem 2. Let \(0<p<1/2\) and \(\varphi \colon \mathbb N\to [1,+\infty)\) be a decreasing function satisfying the condition \(\limsup_{n\to\infty}n^{1/p-2}/\varphi(n)=\infty\). Then there exists a dyadic martingale \(f\in H^p(G)\) such that \(\sup_{n}\| \sigma_n^\kappa(f)/\varphi(n) \| _{L^{p,\infty}}=\infty\).
Remark. In the statement of Theorem 2 \(\sigma_n(f)\) is written instead of \(\sigma_n^\kappa(f)\).