Estimates of Henstock--Kurzweil Poisson integrals

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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 uᵣₚ=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 uᵣₚ=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.