Maximal operator of the Fejér means of triangular partial sums of two-dimensional Walsh–Fourier series

Abstract

It is proved that the maximal operator # of the triangular-Fejér-means of a two-dimensional Walsh-Fourier series is bounded from the dyadic Hardy space H p to L p for all 1=2 < p Ä 1 and, consequently, is of weak type (1,1). As a consequence we obtain that the triangular-Fejér-means 2 n of a function f 2 L 1 converge a.e. to f . The maximal operator # is bounded from the Hardy space H 1=2 to the space weak-L 1=2 and is not bounded from the Hardy space H 1=2 to the space L 1=2 .

Almost Everywhere Convergence of Fejér Means of Two-dimensional Triangular Walsh–Fourier Series

“…These results justify the investigation of the Fejér (or (C, 1)) means of triangular sums of two-dimensional Fourier series defined as (see e.g. [10]):…”

“…The first result in this a.e. convergence issue of triangular means is due to Goginava and Weisz [10]. They proved for the Walsh-Paley system and each integrable function the a.e.…”

Marcinkiewicz-like means of two dimensional Vilenkin--Fourier series

“…The Fejér means of the triangular partial sums of the two-dimensional integrable function f (see e.g. [7]) are…”

Фейеровская Суммируемость По Треугольникам Двойных Рядов Фурье

“…This summability method is rarely investigated in the literature (see the references in [23]). In [12] it is proved that the maximal operator σ △ # := sup n σ △ 2 n f of the Fejér means of the triangular partial sums of the double Walsh-Fourier series is bounded from the dyadic Hardy space H p (G × G) to the L p (G × G) if p > 1/2, is bounded from H 1/2 (G × G) to the space weak-L 1/2 (G × G) and it is not bounded from H 1/2 (G × G) to L 1/2 (G × G).…”