Estimates of Henstock-Kurzweil poisson integrals

• Autores: Erik Talvila
• Localización: Canadian mathematical bulletin, ISSN 0008-4395, Vol. 48, Nº 1, 2005, págs. 133-146
• Idioma: inglés
• DOI: 10.4153/cmb-2005-012-8
• Texto completo no disponible (Saber más ...)
• Resumen

  • If f is a real-valued function on [-pi, pi] that is Henstock-Kurzweil integrable, let ur (theta) be its Poisson integral. It is shown that || ur ||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 ur (theta) is its Poisson integral then for each 1 leq p leq infty the estimate ||ur||p = O((1-r)1/p-1) as r \to 1 is sharp. The Alexiewicz norm estimates || ur || leq || f || (0 leq r < 1) and || ur -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.