Uniform boundedness of sequence of operators associated with the Walsh system and their pointwise convergence (Q6550714)

Let us consider \(\mathbb{I}=[0,1)\) and let \(L_0(\mathbb{I})\) be the set of all a.e finite Lebesgue measurable functions from \(\mathbb{I}\) into \([-\infty,\infty].\) For \(0<p<\infty\) let us denote \(L_p(\mathbb{I})\) as the space of functions \(f\in L_0(\mathbb{I})\), such that, \(\|f\|_p=\left(\int_{\mathbb{I}}|f(x)|^pdx\right)^{1/p}<\infty\) and let \(L_\infty(\mathbb{I})\) denote the set of all functions which satisfy \N\[\N\|f\|_\infty=\{ \{y\in\mathbb{R}/|f(x)|\leq y, \text{for a.e. } x\in\mathbb{I}\}<\infty.\N\]\NNow, let \(x=\sum_{n=0}^{\infty}x_n2^{-(n+1)}\) be the dyadic expansion of \(x\in\mathbb{I}\). For every \(n\in \mathbb{N}\), let \(n=\sum_{k\geq 0}\varepsilon_k(n)2^k\) be the binary expasion, where \(\varepsilon_k(n)=0\) or \(\varepsilon_k(n)=1\) for \(k\in \mathbb{N}\) are called the binary coeficients of \(n\).\N\NThe Rademacher system is defined by \(\rho_n(x)=(-1)^{x_n}\) and the Walsh-Paley system is defined as the sequence of functions defined as, \N\[\N\omega_0(x)=1, \\\N\ \text{and} \\\N\ \omega_n(x)=\prod_{k=0}^{n-1}w_k(x)=(-1)^{\sum_{k=0}^{|n|}\varepsilon_k(n)x_k}, \\\N\ x\in\mathbb{I},\ n\in\mathbb{N}.\N\]\NIn this context, the authors define the Walsh-Dirichlet kernel by \(D_n(x)=\sum_{k=0}^{n-1}w_k(x)\) if \(n\in\mathbb{N}\) and \(D_0(x)=0\). Given a function \(f\in L_1(\mathbb{I})\) its partial sums of the Walsh-Fourier series are defined by \(S_m(f,x)=\sum_{i=0}^{m-1}\hat{f}(i)w_i(x)\), where \(\hat{f}(i)=\int_{\mathbb{I}}f(t)w_i(t)dt\) are the Walsh-Fourier coefficients and particularly, they denote \(E_n(f,x)=S_{2^n}(f,x)\) and \(E^\ast (f,x)=\operatorname*{sup}_{n\in\mathbb{N}}|E_n(f,x)|\). For \(0<p<\infty\), the Hardy space \(H_p(\mathbb{I})\) is the set of all functions \(f\in L_1(\mathbb{I})\), such that, \(\|f\|_{H_p}=\|E^\ast (f)\|_p<\infty.\)\N\NThe Fejer kernel means are defined, respectively, by \(K_n(t)=\frac{1}{n}\sum_{k=1}^{n}D_k(t)\), \(\sigma_n(f,x)=\frac{1}{n}\sum_{k=1}^{n}S_k(f,x)\) and the maximal operators \(\sigma_\ast (f)=\operatorname*{sup}_{n\in\mathbb{N}}|\sigma_n(f)|\) and \(\sigma_\ast ^{a,b,c}(f)=\operatorname*{sup}_{n\in\mathbb{N}}|f\ast|K_n||\).\N\NLet \(\mathbb{T}=(t_{k,n})\) be an infinite triangular matrix satisfying the following conditions: a) \(t_{k,n}\geq 0\), \(k,n\in\mathbb{N}\), b) \(t_{k,n}=0\), \(k>n\), c) \(\sum_{k=1}^n t_{k,n}=1\). Then, the authors introduce the following operators for \(n\in\mathbb{N}\), \N\[\NT_n(f,x)=\sum_{k=1}^n t_{k,n}S_k(f,x), \\\N\ V(n,\mathbb{T})=\sum_{k=1}^{|n|}|\varepsilon_{k+1}(n)-\varepsilon_k(n)|\tau_n^{n^(k)},\N\]\Nwhere \(\tau_n^{(k)}=\sum_{l=k}^nt_{l,n}\) and \(n^{(s)}=\sum_{j=s}^\infty\varepsilon_j(n)2^j\).\N\NTherefore, the results of this article can be summarized as follows. Let \(\{n_a\}_{a\in\mathbb{N}}\) be a subsequence of \(\mathbb{N}\), then the following statements are equivalent\N\begin{itemize}\N\item[1)] The sequence of operators \(\{T_{n_a}\}\) is uniformly bounded from \(H_1(\mathbb{I})\) to \(L_1(\mathbb{I})\).\N\item[2)] The sequence of operators \(\{T_{n_a}\}\) is uniformly bounded from \(L_\infty(\mathbb{I})\) to \(L_\infty(\mathbb{I})\).\N\item[3)] There are constants \(C_1>0\) and \(C_2>0\), such that \N\[\N\operatorname*{sup}_{a\in\mathbb{N}}|T_{n_a}(f)|\leq C_1E^\ast (|f|)+C_2\sigma_\ast ^{a,b,c}(|f|).\N\]\N\item[4)] \(\operatorname*{sup}_{a\in\mathbb{N}} V(n_a,\mathbb{I})<\infty.\)\N\end{itemize}\NAlso, if \(\{T_{n_a}\}\) is uniformly bounded from \(H_1(\mathbb{I})\) to \(L_1(\mathbb{I})\), then \(\lim_{n\to \infty}T_{n_a}(f,x)=f(x),\) for a.e. \(x\in \mathbb{I}\) and each \(f\in L_1(\mathbb{I})\).\N\NFinally, let \(\{n_a\}_{a\in\mathbb{N}}\) be a subsequence of \(\mathbb{N}\) with \(t_{n_a,n_a}=o(1)\), as \(n\to \infty\). Assume that one of the statements of items 1) - 4) is fulfilled, then for each \(f\in L_1(\mathbb{I})\), the sequence \(\{T_{n_a}\}\) converges to \(f\) at every Walsh-Lebesgue point.