The maximal Riesz operator of two-dimensional Fourier transforms and Fourier series on Hp(R x R) and Hp(T x T)

JOURNAL OF APPROXIMATION THEORY

卷:102

Article

Weisz, F

关键词: Hardy spaces;    rectangle p-atom;    atomic decomposition;    interpolation;    Riesz means;   

DOI  :  10.1006/jath.1999.3375

来源: Elsevier

【 摘 要 】

It is proved that the maximal operator of the two-parameter Riesz means with parameters alpha, beta less than or equal to 1 is bounded from L-p(R-2) to L-p(R-2) (1 < p < infinity). The two-dimensional classical Hardy spaces H-p(R x R) are introduced and it is shown that the maximal Riesz operator of a tempered distribution is also bounded from H-p (R x R) to L-p(R-2) (max{1/(alpha + 1), 1/(beta + 1)} < p less than or equal to infinity) and is of weak type i (H-1(#)(R x R), L-1(R-2)) where the Hardy space H-1(#)(R x R) is defined by the hybrid maximal function. As a consequence we obtain that the Riesz means of a function f epsilon H-1(#)(R x R) superset of L log L(R-2) converge a.e. to the function in question. Moreover, wt prove that the Riesz means are uniformly bounded on the spaces H-p(R x R) whenever max { 1/(x + 1), 1/(beta + 1)} < p < infinity. Thus, in case f epsilon H-p(R x R), the Riesz means converge to f in H-p(R x R) norm. The same results are proved for the conjugate: Riesz means and Tor two-parameter Fourier series, too. (C) 2000 Academic Press.

【 授权许可】

Free