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.

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.

Expansions with Poisson kernels and related topics

2010 ◽  
Vol 53 (1) ◽  
pp. 153-173 ◽  
Author(s):  
Cristina Giannotti ◽  
Paolo Manselli
Keyword(s):  
Unit Disc ◽  
Poisson Kernel ◽  
Cauchy Type ◽  
Two Dimensional ◽  
Type Problem ◽  
Special Sequence ◽  
Poisson Kernels ◽  
Cauchy Type Problem ◽  
In Series

AbstractLet P(r, θ) be the two-dimensional Poisson kernel in the unit disc D. It is proved that there exists a special sequence {ak} of points of D which is non-tangentially dense for ∂D and such that any function on ∂D can be expanded in series of P(|ak|, (·)–arg ak) with coefficients depending continuously on f in various classes of functions. The result is used to solve a Cauchy-type problem for Δu = μ, where μ is a measure supported on {ak}.

scholarly journals Linearised Moser-Trudinger inequality

2000 ◽  
Vol 62 (3) ◽  
pp. 445-457 ◽  
Author(s):  
Meelae Kim
Keyword(s):  
Unit Disc ◽  
Higher Dimensions ◽  
Trudinger Inequality ◽  
Limiting Case

As a limiting case of the Sobolev imbedding theorem, the Moser-Trudinger inequality was obtained for functions in with resulting exponential class integrability. Here we prove this inequality again and at the same time get sharper information for the bound. We also generalise the Linearised Moser inequality to higher dimensions, which was first introduced by Beckner for functions on the unit disc. Both of our results are obtained by using the method of Carleson and Chang. The last section introduces an analogue of each inequality for the Laplacian instead of the gradient under some restricted conditions.

scholarly journals Nontangential Limits for Modified Poisson Integrals of Boundary Functions in a Cone

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Lei Qiao
Keyword(s):  
Poisson Integrals ◽  
Nontangential Limits ◽  
Boundary Functions ◽  
Tangential Limits

Our aim in this paper is to deal with non-tangential limits for modified Poisson integrals of boundary functions in a cone, which generalized results obtained by Brundin and Mizuta-Shimomura.

scholarly journals Poisson integrals for standard weighted Laplacians in the unit disc

2013 ◽  
Vol 65 (2) ◽  
pp. 447-486 ◽  
Author(s):  
Anders OLOFSSON ◽  
Jens WITTSTEN
Keyword(s):  
Unit Disc ◽  
Poisson Integrals

scholarly journals On minimal thinness, reduced functions and Green potentials

1993 ◽  
Vol 36 (1) ◽  
pp. 87-106 ◽  
Author(s):  
Matts Essén
Keyword(s):  
Unit Circle ◽  
Recent Work ◽  
Unit Disc ◽  
Minimum Principle ◽  
Higher Dimensions ◽  
Connected Subset ◽  
Whitney Decomposition ◽  
Minimal Thinness ◽  
Image Position ◽  
Green Potentials

Let Ω be an open connected subset of the unit disc U, let E = U\Ω and let {Ωk} be a Whitney decomposition of U. If z(Q) is the centre of the “square” Q, if T is the unit circle and t = dist.(Q, T), we considerwhere Ek = E ∩ Qk and c(Ek) is the capacity of Ek. We prove that the set E is minimally thin at τ ∈ T in U if and only if W(τ)< ∞. We study functions of type W and discuss the relation between certain results of Naim on minimal thinness [15], a minimum principle of Beurling [3], related results due to Dahlberg [7] and Sjögren [16] and recent work of Hayman-Lyons [15] (cf. also Bonsall [4]) and Volberg [19]. For simplicity, we discuss our problems in the unit disc U in the plane. However, the same techniques work for analogous problems in higher dimensions and in more complicated regions.

scholarly journals Vanishing l1-sums of the Poisson kernel, and sums with positive coefficients

1989 ◽  
Vol 32 (3) ◽  
pp. 431-447 ◽  
Author(s):  
F. F. Bonsall ◽  
D. Walsh
Keyword(s):  
Unit Disc ◽  
Poisson Kernel ◽  
Open Unit ◽  
Complex Numbers ◽  
Open Unit Disc ◽  
Image Position ◽  
Positive Coefficients

For z in D and ζ in ∂D, we denote by pz(ζ) the Poisson kernel (1 − │z│2)│1 − z̄ζ−2 for the open unit disc D. We ask for what countable sets {an:n∈ℕ} of points of D there exist complex numbers λn withby which we mean that the series converges to zero in the norm of L1(∂D).

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