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Estimates of Henstock-Kurzweil Poisson integrals. (English) Zbl 1073.26004
Summary: If \(f\) is a real-valued function on \([-\pi,\pi]\) that is Henstock-Kurzweil integrable, let \(u_r(\theta)\) be its Poisson integral. It is shown that \(\| u_r\|_p = o(1/(1 -r))\) as \(r\to 1\) and this estimate is sharp for \(1\leq p\leq \infty\). If \(\mu\) is a finite Borel measure and \(u_r(\theta)\) is its Poisson integral then for each \(1\leq p\leq\infty\) the estimate \(\| u_r\|_p = O((1 - r)^{1/p-1})\) as \(r\to 1\) is sharp. The Alexiewicz norm estimates \(\| u_r\|\leq \| f\|\) \((0\leq r <1)\) and \(\| u_r-f\|\to 0\) \((r\to 1)\) 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.

MSC:
26A39 Denjoy and Perron integrals, other special integrals
31A20 Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions
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