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Estimates of Henstock-Kurzweil Poisson Integrals

Published online by Cambridge University Press:  20 November 2018

Erik Talvila
Erik Talvila*
Affiliation:
Department of Mathematics and Statistics, University College of the Fraser Valley, Abbotsford, BC, V2S 7M8 e-mail: Erik.Talvila@ucfv.ca
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Abstract

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If $f$ is a real-valued function on $\left[ -\pi ,\,\pi \right]$ that is Henstock-Kurzweil integrable, let ${{u}_{r}}(\theta )$ be its Poisson integral. It is shown that ${{\left\| {{u}_{r}} \right\|}_{p}}\,=\,o\left( 1/\left( 1-r \right) \right)$ as $r\,\to \,1$ and this estimate is sharp for $1\,\le \,p\,\le \,\infty $. If $\mu $ is a finite Borel measure and ${{u}_{r}}(\theta )$ is its Poisson integral then for each $1\,\le \,p\,\le \,\infty $ the estimate ${{\left\| {{u}_{r}} \right\|}_{p}}\,=\,O\left( {{\left( 1-r \right)}^{1/p-1}} \right)$ as $r\,\to \,1$ is sharp. The Alexiewicz norm estimates $\left\| {{u}_{r}} \right\|\,\le \,\left\| f \right\|$$\left( 0\,\le \,r\,<\,1 \right)$ and $\left\| {{u}_{r}}-f \right\|\,\to 0\left( r\to 1 \right)$ 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.

Keywords

26A3931A20
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 1 , 01 March 2005 , pp. 133 - 146
Copyright
Copyright © Canadian Mathematical Society 2005

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