Differential operators for a scale of Poisson type kernels in the unit disc

Olofsson, Anders

doi:10.1007/s11854-014-0019-4

Cited by 43 publications

(29 citation statements)

References 12 publications

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“…In [24], Olofsson proved that, for parameter values α > −1, a function f ∈ C 2 (D) satisfies (1.1) if and only if it has the form of a Poisson type integral…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Some properties of a class of elliptic partial differential operators

Chen

Вуоринен

2015

Journal of Mathematical Analysis and Applications

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“…In [24], Olofsson proved that, for parameter values α > −1, a function f ∈ C 2 (D) satisfies (1.1) if and only if it has the form of a Poisson type integral…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Some properties of a class of elliptic partial differential operators

Chen

Вуоринен

2015

Journal of Mathematical Analysis and Applications

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“…Recent work of Olofsson [34] shows the following. We assume that θ is a nonnegative integer for technical reasons; the result may well be true for general real θ > − 1 2 .…”

Section: 2mentioning

confidence: 93%

“…[33]. More recently, in [34] the kernels U N,N are shown to solve the Dirichlet problem in the disk D for a certain (singular) second order elliptic differential operator. It remains to substantiate that U j,N is N-harmonic.…”

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confidence: 99%

Weighted integrability of polyharmonic functions

Borichev

Hedenmalm

2014

Advances in Mathematics

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To address the uniqueness issues associated with the Dirichlet problem for the $N$-harmonic equation on the unit disk $\D$ in the plane, we investigate the $L^p$ integrability of $N$-harmonic functions with respect to the standard weights $(1-|z|^2)^{\alpha}$. The question at hand is the following. If $u$ solves $\Delta^N u=0$ in $\D$, where $\Delta$ stands for the Laplacian, and [\int_\D|u(z)|^p (1-|z|^2)^{\alpha}\diff A(z)<+\infty,] must then $u(z)\equiv0$? Here, $N$ is a positive integer, $\alpha$ is real, and $0\beta(N,p)$ there exist non-trivial functions $u$ with $\Delta^N u=0$ of the given integrability, while for $\alpha\le\beta(N,p)$, only $u(z)\equiv0$ is possible. We also investigate the obstruction to uniqueness for the Dirichlet problem, that is, we study the structure of the functions in $\mathrm{PH}^p_{N,\alpha}(\D)$ when this space is nontrivial. We find a fascinating structural decomposition of the polyharmonic functions -- the cellular (Almansi) expansion -- which decomposes the polyharmonic weighted $L^p$ in a canonical fashion. Corresponding to the cellular expansion is a tiling of part of the $(p,\alpha)$ plane into cells. A particularly interesting collection of cells form the entangled region.Comment: 31 pages, 2 figure

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“…where the function P α is as in (5). Notice that u = P 0 [f ] is the usual Poisson integral representation of a harmonic function in D. The Poisson integral representation u = P α [f ] leads to improved convergence to boundary values (in norm as well as non-tangentially) of solutions of (3) provided the boundary data f ∈ D (T) has some appropriate regularity, see [2][3][4] for results. Let us recall the notion of Lipschitz continuity.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, this more general validity of (8) compared to (7) is quite natural from the point of view of convolution structure in (6) and an interpretation of the differential operator z∂ −z∂ as angular derivative. The growth estimate (8) is established for u = P α [f ] with a fairly economical constant C = C(f ) using a precise result on the L 1 -means of the function P α from Olofsson and Wittsten [2, Section 2] later refined in Olofsson [3] (see Theorem 2.4). We mention that inequality (8) generalizes a recent result by Chen [10,Lemma 3.2].…”

Section: Introductionmentioning

confidence: 99%

Lipschitz continuity for weighted harmonic functions in the unit disc

Olofsson

2019

Complex Variables and Elliptic Equations

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We study membership in Lipschitz classes β (D) for a class of αharmonic functions in the open unit disc D in the complex plane. From earlier work by Olofsson and Wittsten we know that such an α-harmonic function u is the α-harmonic Poisson integral P α [f ] of its boundary value function f on the unit circle T. We determine when the Poisson integral P α [f ] belongs to a Lipschitz class β (D) for the unit disc.

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Differential operators for a scale of Poisson type kernels in the unit disc

Cited by 43 publications

References 12 publications

Some properties of a class of elliptic partial differential operators

Some properties of a class of elliptic partial differential operators

Weighted integrability of polyharmonic functions

Lipschitz continuity for weighted harmonic functions in the unit disc

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