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

The Maximal Riesz, Fej�r, and Ces�ro Operators on Real Hardy Spaces

“…Using an interpolation theorem and the fact that H (1,0) (R 2 ) ⊂ H 1,∞ (R 2 ) (the latter being a Hardy-Lorentz space), Weisz [17] proved the following weak type result: If f ∈ H # 1 (R × R), then the Hilbert transformsf (1,0) ,f (0,1) , andf (1,1) exist a.e. (as ordinary functions) and for all λ > 0 we have H (1,0) on its right-hand side.…”

Several Dimensional θ-Summability and Hardy Spaces

“…We briefly write L p,q or L p,q X d instead of the real Lorentz space L p,q X d , λ (0 < p, q ≤ ∞) and its norm is denoted by · p,q where X = T or IR (for the exact definitions see e. g. Weisz [23] and the references there). .…”

“…It is known that the Riesz means σ α,γ n f → f in L p T 2 norm as n → ∞ without any further condition, provided that f ∈ L p T 2 for some 1 ≤ p < ∞ (see Stein and Weiss [15] and Weisz [23]). It is known that the Riesz means σ α,γ n f → f in L p T 2 norm as n → ∞ without any further condition, provided that f ∈ L p T 2 for some 1 ≤ p < ∞ (see Stein and Weiss [15] and Weisz [23]).…”