On the summmability almost everywhere of the multiple Fourier series at the critical index (Q1064496)

Let \(f(x)=f(x_ 1,x_ 2,...,x_ k)\in L\) on \(Q_ k: -\pi <x_ i\leq \pi\) \((i=1,2,...,k)\) and its Fourier series be \(f(x)\sim \Sigma a_ n e^{inx}\), where \(a_ n=(2\pi)^{-k}\int_{Q_ k}f(x)e^{-inx}dx.\) Then the \(\delta\) th Bochner-Riesz means of the series is \[ (S_ R^{\delta}f)(x)=\sum_{| n| <R}(1-| n|^ 2/R^ 2)^{\delta} a_ n e^{inx}. \] In the present paper, the author establishes the following theorem: If \[ \int_{Q_ k}| f(x)| (\log^+| f(x)|)(\log^+ \log^+ | f(x)|)dx<\infty, \] then \(\lim_{R\to \infty}(S_ R^{\alpha}f)(x)=f(x)\) a.e., where \(\alpha =(k-1)/2\) \((k>1)\).