Norm convergence of double Fejér means on unbounded Vilenkin groups (Q2416486)

Given a sequence $\{m_j\}\subset\mathbb{Z}$, let $M_0=1$ and $M_{k+1}=m_kM_k$. Each $n\in\mathbb{N}$ can be written as a finite sum $n=\sum_{j=0}^\infty n_jM_j$, $n_j\in\mathbb{Z}_{m_j}$. One defines a Vilenkin system $\psi=\{\psi_n\}$ where $\psi_n(x)=\prod_{k=0}^\infty \rho_k^{n_k}(x)$ with $\rho_k(x)=\exp(2\pi i x_k/m_k)$ and $x=(x_0,x_1,\dots)$, $x_j\in \mathbb{Z}_{m_j}$. The Dirichlet kernel is $D_n=\sum_{k=0}^{n-1}\psi_k$ with $D_{M_n}(x)=M_n$ if $x\in I_n$ and $D_{M_n}(x)=0$ if $x\in G\setminus I_n$, where $G=\prod_j \mathbb{Z}_{m_j}$ and $I_n(x)=\{y\in G: y_i=x_i, i=0,1,\dots, n-1\}$. Let $\hat{f}(i,j)=\langle f, \psi_i\psi_j\rangle$ with respect to Haar measure on $G\times G$ and define rectangular partial sums of the Vilenkin-Fourier series $S_{n,m}f(x)=\sum_{i=0}^{n-1}\sum_{j=0}^{m-1} \hat{f}(i,j)\,\psi_i(x)\psi_j(y)$. Corresponding Fejér means are defined by $\sigma_{n,m}(f)=\frac{1}{nm}\sum_{i=0}^{n-1}\sum_{j=0}^{m-1} S_{ij}(x,y;f)$ which can also be expressed by convolution against corresponding Fejér kernels.\par Let $X(G\times G)=C(G\times G)$ or $L^1(G\times G)$. Define moduli of continuity $\omega_1(1/M_n;f)_X=\sup\{\Vert f(\cdot-u,\cdot)-f(\cdot,\cdot)\Vert_X: u\in I_n\}$ and similarly $\omega_2$ by shifting in the second coordinate, and $\omega_{12}(1/M_n,1/M_\ell;f)=\sup\{\Vert f(\cdot-u,\cdot-v)-f(\cdot-u,\cdot)-f(\cdot,\cdot-v)-f(\cdot,\cdot)\Vert_X: u\in I_n, c\in I_\ell\}$. \par The first main result states that if $\omega_1(1/M_k;f)_X=o(1/\log^2(m_k))$ and $\omega_2(1/M_\ell;f)_X=o(1/\log^2(m_\ell))$ then $\Vert \sigma_{n,m}f-f\Vert_X\to 0$ as $n,m\to\infty$. The second main theorem states that if $f\in C(G\times G)$, if $\omega_1(1/M_k;f)_C=O(1/\log^2(m_k))$ and $\omega_2(1/M_\ell;f)_C=O(1/\log^2(m_\ell))$ then $\overline{\lim}_{n\to\infty}\Vert \sigma_{n,m}f-f\Vert_C\to 0$ as $n,m\to\infty$. The third main theorem draws the same conclusion with $C$ replaced by $L^1$. Several lemmas provide suitable bounds on norms of the Fejér kernel $K_n$, for example, $\Vert K_n\Vert_1\leq C\sum_{i=0}^A \frac{\log m_i}{2^{A-i}}(A-i+1)$ with some absolute constant $C$ when $M_A\leq n<M_{A+1}$. The techniques of proof involve estimates over lacunary annular type regions and mixed moduli of continuity bounds allow in turn bounds for double convolutions.