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

Bibliographie

[1] [1] Axler, S., Bourdon, P. and Ramey, W., Harmonic function theory. New York, Springer-Verlag, 2001. Google Scholar

[2] [2] Benedicks, M. and Pfeffer, W. F., The Dirichlet problem with Denjoy-Perron integrable boundary condition. Canad. Math. Bull. 28(1985), 113–119. Google Scholar

[3] [3] Čelidze, V. G. and Džvaršeĭšvili, A. G., The theory of the Denjoy integral and some applications. (trans. Bullen, P. S.), Singapore, World Scientific, 1989. Google Scholar

[4] [4] Folland, G. B., Real analysis. New York, Wiley, 1999. Google Scholar

[5] [5] Gradshteyn, I. S. and Ryzhik, I. M., Table of integrals, series, and products, (trans. Scripta Technica, ed. Jeffrey, A.), San Diego, Academic Press, 2000. Google Scholar

[6] [6] Shapiro, V. L., The uniqueness of functions harmonic in the interior of the unit disc. Proc. London Math. Soc. 13(1963), 639–652. Google Scholar

[7] [7] Swartz, C., An introduction to functional analysis. New York, Marcel Dekker, 1992. Google Scholar

[8] [8] Swartz, C., Introduction to gauge integrals. Singapore, World Scientific, 2001. Google Scholar

[9] [9] Talvila, E., Henstock-Kurzweil Fourier transforms. Illinois J. Math. 46(2002), 1207–1226. Google Scholar

[10] [10] Talvila, E., Continuity in the Alexiewicz norm, (to appear). Google Scholar

[11] [11] Wolf, F., The Poisson integral. A study in the uniqueness of harmonic functions. Acta. Math. 74(1941), 65–100. Google Scholar