Characterizations of the harmonic Hardy space h1 on the real ball

We prove some characterizations of the space h1 and use them to give new proofs of a theorem of Zygmund and a theorem of Kologorov and Smirnov.

Estimates of Henstock-Kurzweil Poisson Integrals

If f is a real-valued function on [−π, π] that is Henstock-Kurzweil integrable, let ur(θ) be its Poisson integral. It is shown that ∥ur∥p = o(1/(1 − r)) as r → 1 and this estimate is sharp for 1 ≤ p ≤ ∞. If μ is a finite Borel measure and ur(θ) is its Poisson integral then for each 1 ≤ p ≤ ∞ the estimate ∥ur∥p = O((1−r)1/p−1) as r → 1 is sharp. The Alexiewicz norm estimates ∥ur∥ ≤ ∥f ∥ (0 ≤ r < 1) and ∥ur − f∥ → 0 (r → 1) hold. These estimates lead to two uniqueness theorems for the Dirichlet problem in the unit disc with Henstock-Kurzweil integrable boundary data. There are similar growth estimates when u is in the harmonic Hardy space associated with the Alexiewicz norm and when f is of bounded variation.