Triangular Cesàro summability of two dimensional Fourier series
The paper is devoted to the triangular summability of Fourier series, more precisely, the author shows that the maximal operator of the triangular \((C,\alpha)\) Cesàro means of a two-dimensional Fourier series is bounded from the periodic Hardy space \(H_p(\mathbb T^2)\) to \(L_p(\mathbb T^2)\) for all \(p\) from the range \(2/(2+\alpha)<p\leq\infty\) and, consequently, is of weak type \((1,1)\). The proof is very much two-dimensional, and it is not clear what may happen in higher dimension. As a consequence, an old Herriot's result (proved in a different way) follows as the triangular Cesàro means converge almost everywhere. It is worth mentioning that triangular summability has a longer history than that of which the reader may get an impression from the paper. We give only two absolutely necessary additional references: \textit{O. I. Kuznetsova} [Sib.\ Mat.\ Zh.\ 18, 629--636 (1977; Zbl 0364.42018)] and \textit{A. N. Podkorytov} [Vestn.\ Leningr.\ Univ., Math.\ 11, 197--205 (1983; Zbl 0517.42020)].
scientific article; zbMATH DE number 5990617
scientific article