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

Abstract

We prove that the maximal Riesz operator σ α,γ * is of strong type from L 1 (R) ∩ H p (R) to L p (R) for α, γ > 0 and 1/(1 + α) < p ≤ 1, it is of weak type for α, γ > 0 and 1/(1 + α) = p, and these results are best possible. The proofs are based on sharp estimates of the derivatives of the Riesz kernel. We characterize the real Hardy space H p (R) in terms of σ α,1 * for 1/(1 + α) < p ≤ 1, and draw consequences for real Hardy spaces on R 2 , as well. For example, an integrable function f belongs to H 1 (R) if and only if the maximal Fejér operator σ 1,1* applied to f belongs to L 1 (R). We also establish analogous results for real Hardy spaces on T and T 2 .It is well known that . p is a norm if 1 ≤ p ≤ ∞ and is a quasinorm if 0 < p < 1, and each of the spaces L p (R) is complete with respect to . p .Math Subject Classifications. primary 42A24, 42A38, 42A50, 42B08; secondary 46F10, 47B38.