Generalized localization and equiconvergence of expansions in double trigonometric series and in the Fourier integral for functions from \(L(\ln^+ L)^2\) (Q1387342)

Among others, the following theorems are announced without proofs. Theorem 1. If \(\Omega\subset\mathbb{T}^2\) is a nonempty open set, \(f\in L(\ln^+ L)^2(\mathbb{T}^2)\), and \(f(x)= 0\) on \(\Omega\), then \[ \lim_{n\to\infty} S_n(x; f)= 0\quad\text{for almost all }x\in\Omega, \] where \(S_n(x; f)\) denotes the rectangular sum of the double trigonometric Fourier series of \(f\). Furthermore, for every \(\varepsilon> 0\) there exist an open set \(\Omega_\varepsilon\subset \Omega\) such that \(\mu\Omega_\varepsilon> \mu\Omega- \varepsilon\) (where \(\mu\) denotes the two-dimensional Lebesgue measure) and a constant \(C_\varepsilon> 0\) such that \[ \int_{\Omega_\varepsilon} \Biggl( \sup_{n\in\mathbb{Z}^2_0} | S_n(x; f)|\Biggr) dx\leq C_\varepsilon \int_{\mathbb{T}^2}| f(x)|(\ln^+| f(x)|)^2 dx+ C_\varepsilon. \] Theorem 2. For every measurable set \(\Omega\subset\mathbb{T}^2\) such that \(\mu\Omega> 0\) and \(\Omega\) is not dense in \(\mathbb{T}^2\), there exist a subset \(\Omega_0\subset\Omega\) with \(\mu\Omega_0> 0\) and a function \(f\in L(\ln^+\ln^+ L)^{1- \varepsilon}(\mathbb{T}^2)\), where \(0<\varepsilon< 1\), such that \(f(x)= 0\) on \(\Omega\) and \[ \limsup_{n\to\infty} | S_n(x; f)|= +\infty\quad\text{for almost all }x\in\Omega_0. \] {}.