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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.

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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