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Estimates of Henstock--Kurzweil Poisson integrals

Erik Talvila
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Abstract

If $f$ is a real-valued function on $[-π,π]$ that is Henstock--Kurzweil integrable, let $u_r(θ)$ 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 $μ$ is a finite Borel measure and $u_r(θ)$ 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.

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Erik Talvila. 2004-06-18. Estimates of Henstock--Kurzweil Poisson integrals. https://arxiv.org/abs/math/0406371

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