Estimates of Henstock--Kurzweil Poisson integrals

2004/06/18 by Erik Talvila, Talvila, Erik
Mathematics · #26A39 #31A20 #Algebraic and Geometric Analysis #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials #math.AP #math.CA #msc:26A39 #msc:31A20

paper · pdf · doi:10.48550/arxiv.math/0406371

To appear in Canadian Mathematical Bulletin

arxiv created 2004/06/18 · openalex publication_date 2004/06/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

If f is a real-valued function on [-π,π] that is Henstock--Kurzweil integrable, let u_r(θ) be its Poisson integral. It is shown that ‖u_r‖_p=o(1/(1-r)) as r→ 1 and this estimate is sharp for 1≤ p≤∞. If μ is a finite Borel measure and u_r(θ) is its Poisson integral then for each 1≤ p≤ ∞ the estimate ‖u_r‖_p=O((1-r)^1/p-1) as r→ 1 is sharp. The Alexiewicz norm estimates ‖u_r‖≤‖f‖ (0≤ r<1) and ‖u_r-f‖→ 0 (r→ 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.

Related

• Continuity in the Alexiewicz norm
• A product convergence theorem for Henstock--Kurzweil integrals
• Estimates of the remainder in Taylor's theorem using the Henstock--Kurzweil integral
• The Fourier transform with Henstock--Kurzweil and continuous primitive integrals
• A convergence theorem for ap-Henstock-Kurzweil integral and its relation to topology