Estimates of Henstock--Kurzweil Poisson integrals

Abstract

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 $\Vert u_r{\Vert }_p=o(1/(1-r))$ 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 $\Vert u_r{\Vert }_p=O((1-r{)}^{1/p-1})$ as $r\to 1$ is sharp. The Alexiewicz norm estimates $\Vert u_r\Vert \le \Vert f\Vert$ ($0\le r<1$) and $\Vert u_r-f\Vert \to 0$ ($r\to 1$) 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.

To appear in Canadian Mathematical Bulletin