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Estimates of HenstockKurzweil Poisson Integrals


Published:20050301
Printed: Mar 2005
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Abstract
If $f$ is a realvalued function on $[\pi,\pi]$ that
is HenstockKurzweil integrable, let $u_r(\theta)$ be its Poisson
integral. It is shown that $\u_r\_p=o(1/(1r))$ 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
$\u_r\_p=O((1r)^{1/p1})$ as $r\to 1$ is sharp.
The Alexiewicz
norm estimates $\u_r\\leq\f\$ ($0\leq r<1$) and $\u_rf\\to 0$
($r\to 1$) hold. These estimates lead to two uniqueness theorems for
the Dirichlet problem
in the unit disc with HenstockKurzweil 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.