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1. CMB 2005 (vol 48 pp. 133)
Estimates of Henstock-Kurzweil Poisson Integrals 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.
Categories:26A39, 31A20 |