On Marcinkiewicz-Riesz Summability of Fourier Integrals in Hardy-Spaces

✍ Scribed by P. Oswald

• Publisher: John Wiley and Sons
• Year: 1987
• Tongue: English
• Weight: 614 KB
• Volume: 133
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19871330112

✦ Synopsis

In this paper we intend to prove weak type estimates for maximal functions related to a.e. summability of FOURIER integrals in the spaces Ifp( R N ) where O -= p ~l and N>1.

Let

( 1 ) &!?(z)=(2n)-" f (1-(& -lyIJ2): $f(y) eiz%zg, &>O , RN be the MARCINKIEWICZ-RIESZ means of order 6zO of the function (or distribution) f ( z ) defined on R N . Here and in the following $f(y) = f f ( x )e-'zYdz denotes the FOURIER transform of f(z), lyl-= max lyil is the sup-norm while [yI stands i-1, ...,N for the euclidean norm of y = (yl, ..., y ~) E R N . These means are one of the natural generalizations of the one-dimensional CESARO -RIESZ means to multiple FOURIER integrals (and series) and were introduced by MARCIXKIEWICZ [7] in the case of double FOURIER series. Another important generalization goes back to BOCHNER [2] and leads to the BOCHNER-RIESZ means (2) Roughly speaking, while (2) represents spherical RIESZ summation, in (1) we have RIESZ summation applied to the partial sums over cubes of the FOUBIEB integral .

There. exists an extensive literature on properties of the means (1) and (2) and their periodic, counterparts (especially for the Lp spaces, 1 S p S-), we refer to the book [14] and the review articles [l], 151, [20] where further references could be found.

Here we are interested in the case of functions f(z) belonging to, the HARDY space H p ( R N ) for some$ s 1 and the problem of a.e. convergence of the means (1). Concerning the analogous question for the BOCHNER-RIESZ means (2) STEIN, RN S:/(x) = ( 2 n ) -~ J (I -( E -lz~1)2)$8/(y) -ei2%y, E -o . RN