On the relationship between rectangular convergence and spherical convergence of multiple Haar series (Q2892029)

Let \(n\geq 2\), \({\mathbf I}^n= [0,1]^n\), and let \(\{h_k\}_{k\in\mathbb{N}^n}\) be the \(n\)-multiple Haar system. For a given \(n\)-tuple Haar series NEWLINE\[NEWLINE\sigma= \sum c_k h_kNEWLINE\]NEWLINE and for \(x\in{\mathbf I}^n\), \(m\in\mathbb{N}^n\), and \(r>0\), denote by \(S_m(\sigma)(x)\), \(S_r(\sigma)(x)\), \(G_m(\sigma)(x)\), and \(G_r(\sigma)(x)\) the corresponding rectangular partial sum, spherical partial sum, rectangular general term, spherical general term.NEWLINENEWLINE The author announces the following three theorems without proofs.NEWLINENEWLINE Theorem 1. If \(\sigma\) converges rectangularly at a dyadic-irrational point \(x\) and \(G_m(\sigma) (x)\) converges to 0 strongly, then \(\sigma\) converges spherically at \(x\).NEWLINENEWLINE Theorem 2. Let \(f\in L(\log^+L)^{n-2}({\mathbf I}^n)\). Then every \((n-1)\)-dimensional section of the sequence \(\{G_m(\sigma)(x)\}\) strongly converges to 0, for a.e. \(x\in{\mathbf I}^n\).NEWLINENEWLINE Theorem 3. For every \(n\geq 3\) and every \(f\in L\setminus L(\log^+L)^{n-2}\), there exists a function \(g\) equi-measurable with \(f\) such that NEWLINE\[NEWLINE\lim_{m\to\infty}\,S_m(g)(x)= g(x)\quad\text{and}\quad \limsup_{r\to\infty} |G_r(g)(x)|= \inftyNEWLINE\]NEWLINE for almost all \(x\in{\mathbf I}^n\).