Estimates of Henstock-Kurzweil Poisson Integrals

Talvila, Erik

Canadian mathematical bulletin, Tome 48 (2005) no. 1, pp. 133-146

Résumé

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.

Détail

DOI : 10.4153/CMB-2005-012-8

Mots-clés : 26A39, 31A20