The comparison principle and Dirichlet problem in the class , p > 0
Pham Hoang Hiep (2006)
Annales Polonici Mathematici
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We establish the comparison principle in the class . The result obtained is applied to the Dirichlet problem in .
Pham Hoang Hiep (2006)
Annales Polonici Mathematici
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We establish the comparison principle in the class . The result obtained is applied to the Dirichlet problem in .
Rafał Czyż (2010)
Annales Polonici Mathematici
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Let Ω be a bounded hyperconvex domain in ℂn and let μ be a positive and finite measure which vanishes on all pluripolar subsets of Ω. We prove that for every continuous and strictly increasing function χ:(-∞,0) → (-∞,0) there exists a negative plurisubharmonic function u which solves the Monge-Ampère type equation . Under some additional assumption the solution u is uniquely determined.
Sławomir Kołodziej (2003)
Annales Polonici Mathematici
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regularity of the solutions of the complex Monge-Ampère equation in ℂPⁿ with the n-root of the right hand side in is proved.
R. Balasubramanian, B. Calado, H. Queffélec (2006)
Studia Mathematica
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We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if with , then and even slightly better, and , C being an absolute constant.
Małgorzata Michalska (2011)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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In this paper we discuss characterizations of Dirichlet type spaces on the unit ball of obtained by P. Hu and W. Zhang [2], and S. Li [4].
Mohamed Ben Chrouda (2016)
Annales Polonici Mathematici
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This paper deals with the questions of the existence and uniqueness of a solution to the Dirichlet problem associated with the Dunkl Laplacian as well as the hypoellipticity of on noninvariant open sets.
Daniel Carando, Andreas Defant, Domingo A. Garcí, Manuel Maestre, Pablo Sevilla-Peris (2015)
Acta Arithmetica
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Denote by Ω(n) the number of prime divisors of n ∈ ℕ (counted with multiplicities). For x∈ ℕ define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial we have . We prove that the asymptotically correct order of L(x) is . Following Bohr’s vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows translating various results...
B. N. Rachajsky (1969)
Matematički Vesnik
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Slimane Benelkourchi, Vincent Guedj, Ahmed Zeriahi (2008)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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Let be a compact Kähler manifold and be a smooth closed form of bidegree which is nonnegative and big. We study the classes of -plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class . This is done by...
Mohamad Charabati (2015)
Annales Polonici Mathematici
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We consider the Dirichlet problem for the complex Monge-Ampère equation in a bounded strongly hyperconvex Lipschitz domain in ℂⁿ. We first give a sharp estimate on the modulus of continuity of the solution when the boundary data is continuous and the right hand side has a continuous density. Then we consider the case when the boundary value function is and the right hand side has a density in for some p > 1, and prove the Hölder continuity of the solution.
Pham Hoang Hiep (2007)
Annales Polonici Mathematici
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We study boundary values of functions in Cegrell’s class .
Manfred Denker, Atsushi Imai, Susanne Koch (2007)
Colloquium Mathematicae
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We define thin equivalence relations ∼ on shift spaces and derive Dirichlet forms on the quotient space in terms of the nearest neighbour averaging operator. We identify the associated Laplace operator. The conditions are applied to some non-self-similar extensions of the Sierpiński gasket.
Shi-An Han, Ze-Hua Zhou (2016)
Czechoslovak Mathematical Journal
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We investigate isometric composition operators on the weighted Dirichlet space with standard weights , . The main technique used comes from Martín and Vukotić who completely characterized the isometric composition operators on the classical Dirichlet space . We solve some of these but not in general. We also investigate the situation when is equipped with another equivalent norm.
Jean-Pierre Demailly, Sławomir Dinew, Vincent Guedj, Pham Hoang Hiep, Sławomir Kołodziej, Ahmed Zeriahi (2014)
Journal of the European Mathematical Society
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Let be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on with right hand side, . The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range of the complex Monge-Ampère operator acting on -plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with -density belong to and proving that...
Ertan Elma (2021)
Czechoslovak Mathematical Journal
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Let be a nonprincipal Dirichlet character modulo a prime number and let . Define the mean value We give an identity for which, in particular, shows that for fixed and .
Marek Fila, Michael Winkler (2008)
Journal of the European Mathematical Society
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We consider a reaction-diffusion-convection equation on the halfline (0,1) with the zero Dirichlet boundary condition at . We find a positive selfsimilar solution which blows up in a finite time at while remains bounded for .
Per Åhag, Rafał Czyż, Pham Hoàng Hiêp (2007)
Annales Polonici Mathematici
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The energy class is studied for 0 < p < 1. A characterization of certain bounded plurisubharmonic functions in terms of and its pluricomplex p-energy is proved.
Urban Cegrell (2012)
Annales Polonici Mathematici
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For μ a positive measure, we estimate the pluricomplex potential of μ, , where g(x,y) is the pluricomplex Green function (relative to Ω) with pole at y.
Eric Bedford (1985)
Annales Polonici Mathematici
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Per Åhag, Rafał Czyż (2007)
Annales Polonici Mathematici
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Let be a bounded hyperconvex domain in and set , j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.
Urban Cegrell (2007)
Annales Polonici Mathematici
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We show that if a decreasing sequence of subharmonic functions converges to a function in then the convergence is in .
Niraj Kumar, Garima Manocha (2018)
Mathematica Bohemica
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Let be a class of entire functions represented by Dirichlet series with complex frequencies for which is bounded. Then is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. is also proved to be a total set. Conditions for the existence of inverse, topological zero divisor and continuous linear functional for any element belonging to have also been established.
Vladimir A. Kondratiev, Olga A. Oleinik (1986)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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Per ogni soluzione della (1) nel dominio limitato ,, appartenente a e soddisfacente le condizioni (2), si dimostra la maggiorazione (5), valida nell'intorno di ogni punto del contorno; si consente a di essere singolare in .
Jerzy Kalina (1983)
Annales Polonici Mathematici
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Sophie Grivaux (2013)
Studia Mathematica
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If is a strictly increasing sequence of integers, a continuous probability measure σ on the unit circle is said to be IP-Dirichlet with respect to if as F runs over all non-empty finite subsets F of ℕ and the minimum of F tends to infinity. IP-Dirichlet measures and their connections with IP-rigid dynamical systems have recently been investigated by Aaronson, Hosseini and Lemańczyk. We simplify and generalize some of their results, using an approach involving generalized Riesz...