Cesàro Summability of Two-Parameter Trigonometric-Fourier Series

✍ Scribed by Ferenc Weisz

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 325 KB
• Volume: 90
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.3070

✦ Synopsis

The two-dimensional classical Hardy spaces H p (T\_T) on the bidisc are introduced and it is shown that the maximal operator of the CesaÁ ro means of a distribution is bounded from H p (T\_T) to L p (T 2 ) ( 3Â4 (T\_T), L 1 (T 2 )) where the Hardy space H 1 > (T\_T) is defined by the hybrid maximal function. As a consequence we obtain that the CesaÁ ro means of a function f # H 1 > (T\_T)#L log L(T 2 ) converge a.e. to the function in question.
1997 Academic Press

1. Introduction

For double trigonometric Fourier series Marcinkievicz and Zygmund [14] proved that the CesaÁ ro means \_ n, m f of a function f # L 1 (T 2 ) converge a.e. to f as n, m Ä , provided that the pairs (n, m) are in a positive cone, i.e., provided that 2 &$ nÂm 2 $ for any $ 0. A new proof of this result was given by the author [19]. Moreover, Zygmund [24] verified that if f # L log L(T 2 ) then the two-parameter CesaÁ ro summability holds.
We proved in [20] and [19] that, in the one-dimensional case, the maximal operator of the CesaÁ ro means of a distribution is bounded from the Hardy Lorentz space H p, q (T) to L p, q (T) if 3Â4< p , 0<q , and that, in the two-dimensional case, it is bounded from H p, q (T 2 ) ({H p, q (T\_T)) to L p, q (T 2 ) if 5Â6< p , 0<q , provided that the supremum in the maximal operator is taken over a positive cone.
In this paper we generalize these results for the unrestricted maximal operator of the two-parameter trigonometric Fourier series. The analogous result for a two-parameter Walsh Fourier series has been shown by the article no. AT963070 30