The Maximal Riesz Operator of Two-Dimensional Fourier Transforms and Fourier Series on Hp(R×R) and Hp(T×T)

Abstract

It is proved that the maximal operator of the two-parameter Riesz means with parameters α, β⩽1 is bounded from Lp(R2) to Lp(R2) (1<p<∞). The two-dimensional classical Hardy spaces Hp(R×R) are introduced and it is shown that the maximal Riesz operator of a tempered distribution is also bounded from Hp(R×R) to Lp(R2) (max{1/(α+1), 1/(β+1)}<p⩽∞) and is of weak type (H♯1(R×R), L1(R2)) where the Hardy space H♯1(R×R) is defined by the hybrid maximal function. As a consequence we obtain that the Riesz means of a function f∈H♯1(R×R)⊃LlogL(R2) converge a.e. to the function in question. Moreover, we prove that the Riesz means are uniformly bounded on the spaces Hp(R×R) whenever max{1/(α+1), 1/(β+1)}<p<∞. Thus, in case f∈Hp(R×R), the Riesz means converge to f in Hp(R×R) norm. The same results are proved for the conjugate Riesz means and for two-parameter Fourier series, too.