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1
Content available remote

The boundary Harnack principle for the fractional Laplacian

100%
Bogdan K.
Studia Mathematica
|
1997
|
tom 123
|
nr 1
43-80
EN
We study nonnegative functions which are harmonic on a Lipschitz domain with respect to symmetric stable processes. We prove that if two such functions vanish continuously outside the domain near a part of its boundary, then their ratio is bounded near this part of the boundary.
2
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Harmonic functions and Hardy spaces on trees with boundaries

100%
Pytlik T.
Colloquium Mathematicum
|
1992
|
tom 63
|
nr 2
263-272
3
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On the Hartogs-type series for harmonic functions on Hartogs domains in $ℝ^n × ℝ^m$, m ≥ 2

100%
Ligocka E.
Annales Polonici Mathematici
|
1999
|
tom 71
|
nr 2
151-160
EN
We study series expansions for harmonic functions analogous to Hartogs series for holomorphic functions. We apply them to study conjugate harmonic functions and the space of square integrable harmonic functions.
4
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Harmonic functions in a cylinder with normal derivatives vanishing on the boundary

100%
Miyamoto I., Yoshida H.
Annales Polonici Mathematici
|
2000
|
tom 74
|
nr 1
229-235
EN
A harmonic function in a cylinder with the normal derivative vanishing on the boundary is expanded into an infinite sum of certain fundamental harmonic functions. The growth condition under which it is reduced to a finite sum of them is given.
5
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Harmonic functions on the real hyperbolic ball I: Boundary values and atomic decomposition of Hardy spaces

88%
Jaming P.
Colloquium Mathematicum
|
1999
|
tom 80
|
nr 1
63-82
EN
We study harmonic functions for the Laplace-𝔹eltrami operator on the real hyperbolic space $𝔹_n$. We obtain necessary and sufficient conditions for these functions and their normal derivatives to have a boundary distribution. In doing so, we consider different behaviors of hyperbolic harmonic functions according to the parity of the dimension of the hyperbolic ball $𝔹_n$. We then study the Hardy spaces $H^p(𝔹_n)$, 0
6
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"Counterexamples" to the harmonic Liouville theorem and harmonic functions with zero nontangential limits

75%
Bonilla A.
Colloquium Mathematicum
|
2000
|
tom 83
|
nr 2
155-160
EN
We prove that, if μ>0, then there exists a linear manifold M of harmonic functions in $ℝ^N$ which is dense in the space of all harmonic functions in $ℝ^N$ and lim_{{‖x‖→∞} {x ∈ S}} ‖x‖^{μ}D^{α}v(x) = 0 for every v ∈ M and multi-index α, where S denotes any hyperplane strip. Moreover, every nonnull function in M is universal. In particular, if μ ≥ N+1, then every function v ∈ M satisfies ∫_H vdλ =0 for every (N-1)-dimensional hyperplane H, where λ denotes the (N-1)-dimensional Lebesgue measure. On the other hand, we prove that there exists a linear manifold M of harmonic functions in the unit ball 𝔹 of $ℝ^N$, which is dense in the space of all harmonic functions and each function in M has zero nontangential limit at every point of the boundary of 𝔹.
7
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The generalized Neumann-Poincaré operator and its spectrum

41%
Partyka D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Introduction..........................................................................................................................................................................5 Preliminaries. Complex harmonic functions..........................................................................................................................7 I. Spectral values and eigenvalues of a Jordan curve........................................................................................................19  1.1. On a boundary integral..............................................................................................................................................20  1.2. The generalized Cauchy singular integral operator $C_𝕍$.......................................................................................23  1.3. The Hilbert transformation $T_Ω$.............................................................................................................................28  1.4. The boundary space Ḣ²(∂Ω)......................................................................................................................................31  1.5. The generalized Neumann-Poincaré operator $N_𝕍$...............................................................................................36 II. Quasisymmetric automorphisms of the unit circle...........................................................................................................41  2.1. The Douady-Earle extension $E_γ$..........................................................................................................................42  2.2. On an approximation of the Hersch-Pfluger distortion function $Φ_K$......................................................................46  2.3. On the maximal dilatation of the Douady-Earle extension..........................................................................................48  2.4. The Hilbert space H...................................................................................................................................................54  2.5. The linear operator $B_γ$.........................................................................................................................................60 III. The generalized harmonic conjugation operator............................................................................................................64  3.1. The generalized harmonic conjugation operator $A_γ$.............................................................................................64  3.2. Spectral values and eigenvalues of a quasisymmetric automorphism of the unit circle..............................................73  3.3. The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle..............................................80  3.4. Limiting properties of spectral values and eigenvalues of a quasisymmetric automorphism of the unit circle............84 IV. Spectral values of a quasicircle.....................................................................................................................................90  4.1. Characterizations of the boundary space Ḣ²(∂Ω).......................................................................................................91  4.2. Spaces symmetric with respect to a Jordan curve.....................................................................................................93  4.3. Plemelj's formula for a quasicircle..............................................................................................................................96  4.4. The main spectral theorem for quasicircles.............................................................................................................103  4.5. Spectral values and eigenvalues of a quasicircle....................................................................................................108 Appendix. The inner completion of pseudo-normed spaces............................................................................................114 References......................................................................................................................................................................117 List of symbols.................................................................................................................................................................122 Index................................................................................................................................................................................124
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