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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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