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p Harmonic Measure in Simply Connected Domains

John L. Lewis, Kaj Nyström, Pietro Poggi-Corradini (2011)

Annales de l’institut Fourier

Let Ω be a bounded simply connected domain in the complex plane, . Let N be a neighborhood of Ω , let p be fixed, 1 < p < , and let u ^ be a positive weak solution to the p Laplace equation in Ω N . Assume that u ^ has zero boundary values on Ω in the Sobolev sense and extend u ^ to N Ω by putting u ^ 0 on N Ω . Then there exists a positive finite Borel measure μ ^ on with support contained in Ω and such that | u ^ | p - 2 u ^ , φ d A = - φ d μ ^ whenever φ C 0 ( N ) . If p = 2 and if u ^ is the Green function for Ω with pole at x Ω N ¯ then the measure μ ^ coincides with harmonic measure...

p -harmonic measure is not additive on null sets

José G. Llorente, Juan J. Manfredi, Jang-Mei Wu (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

When 1 < p < and p 2 the p -harmonic measure on the boundary of the half plane + 2 is not additive on null sets. In fact, there are finitely many sets E 1 , E 2 ,..., E κ in , of p -harmonic measure zero, such that E 1 E 2 . . . E κ = .

Painlevé's problem and analytic capacity.

Xavier Tolsa (2006)

Collectanea Mathematica

In this paper we survey some recent results in connection with the so called Painlevé's problem and the semiadditivity of analytic capacity γ. In particular, we give the detailed proof of the semiadditivity of the capacity γ+, and we show almost completely all the arguments for the proof of the comparability between γ and γ+.

Parabolic potentials and wavelet transforms with the generalized translation

Ilham A. Aliev, Boris Rubin (2001)

Studia Mathematica

Parabolic wavelet transforms associated with the singular heat operators - Δ γ + / t and I - Δ γ + / t , where Δ γ = k = 1 n ² / x ² k + ( 2 γ / x ) / x , are introduced. These transforms are defined in terms of the relevant generalized translation operator. An analogue of the Calderón reproducing formula is established. New inversion formulas are obtained for generalized parabolic potentials representing negative powers of the singular heat operators.

Perfect sets of finite class without the extension property

A. Goncharov (1997)

Studia Mathematica

We prove that generalized Cantor sets of class α, α ≠ 2 have the extension property iff α < 2. Thus belonging of a compact set K to some finite class α cannot be a characterization for the existence of an extension operator. The result has some interconnection with potential theory.

Periodic harmonic functions on lattices and points count in positive characteristic

Mikhail Zaidenberg (2009)

Open Mathematics

This survey deals with pluri-periodic harmonic functions on lattices with values in a field of positive characteristic. We mention, as a motivation, the game “Lights Out” following the work of Sutner [20], Goldwasser- Klostermeyer-Ware [5], Barua-Ramakrishnan-Sarkar [2, 19], Hunzikel-Machiavello-Park [12] e.a.; see also [22, 23] for a more detailed account. Our approach uses harmonic analysis and algebraic geometry over a field of positive characteristic.

Perron-Frobenius operators and the Klein-Gordon equation

Francisco Canto-Martín, Håkan Hedenmalm, Alfonso Montes-Rodríguez (2014)

Journal of the European Mathematical Society

For a smooth curve Γ and a set Λ in the plane 2 , let A C ( Γ ; Λ ) be the space of finite Borel measures in the plane supported on Γ , absolutely continuous with respect to the arc length and whose Fourier transform vanishes on Λ . Following [12], we say that ( Γ , Λ ) is a Heisenberg uniqueness pair if A C ( Γ ; Λ ) = { 0 } . In the context of a hyperbola Γ , the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Λ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the...

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