The Poisson Kernel and Poisson Integrals

2016 ◽  
pp. 59-81
Author(s):  
Manfred Stoll
Keyword(s):  
Poisson Kernel ◽  
Poisson Integrals

scholarly journals Approach regions for the square root of the Poisson kernel and bounded functions

1997 ◽  
Vol 55 (3) ◽  
pp. 521-527 ◽  
Author(s):  
P. Sjögren
Keyword(s):  
Unit Disc ◽  
Poisson Kernel ◽  
Monotone Function ◽  
Square Root ◽  
Almost Everywhere Convergence ◽  
Poisson Integrals ◽  
Almost Everywhere ◽  
Approach Region ◽  
Boundary Functions ◽  
Bounded Functions

If the Poisson kernel of the unit disc is replaced by its square root, it is known that normalised Poisson integrals of Lp boundary functions converge almost everywhere at the boundary, along approach regions wider than the ordinary non-tangential cones. The sharp approach region, defined by means of a monotone function, increases with p. We make this picture complete by determining along which approach regions one has almost everywhere convergence for L∞ boundary functions.

scholarly journals Approach regions for $l^{p}$ potentials with respect to the square root of the Poisson kernel

2005 ◽  
Vol 96 (2) ◽  
pp. 243
Author(s):  
Martin Brundin
Keyword(s):  
Unit Disc ◽  
Poisson Kernel ◽  
Higher Dimensions ◽  
Square Root ◽  
Poisson Integrals ◽  
Boundary Functions

{If} one replaces the Poisson kernel of the unit disc by its square root, then normalised Poisson integrals of $L^{p}$ boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning ($1\leq p<\infty$) and Sjögren ($p=1$ and $p=\infty$). In this paper we present new and simplified proofs of these results. We also generalise the $L^{\infty}$ result to higher dimensions.

Über das Randverhalten des POISSON-Integrals der polyharmonischen Gleichung

1980 ◽  
Vol 95 (1) ◽  
pp. 157-164 ◽  
Author(s):  
Luis Gonzáles ◽  
Eckart Keller ◽  
Günther Wildenhain
Keyword(s):  
Poisson Integrals

scholarly journals The Failure of Fatou's Theorem on Poisson Integrals of Pettis Integrable Functions

1998 ◽  
Vol 160 (1) ◽  
pp. 28-41 ◽  
Author(s):  
Francisco J Freniche ◽  
Juan Carlos Garcı́a-Vázquez ◽  
Luis Rodrı́guez-Piazza
Keyword(s):  
Poisson Integrals ◽  
Integrable Functions ◽  
Fatou’S Theorem
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