Duality on vector-valued weighted harmonic Bergman spaces

Salvador Pérez-Esteva

Displaying similar documents to “Duality on vector-valued weighted harmonic Bergman spaces”

On Bell's duality theorem for harmonic functions

Joaquín Motos, Salvador Pérez-Esteva (1999)

Studia Mathematica

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Define $h^{\infty} (E)$ as the subspace of $C^{\infty} (B ̅ L, E)$ consisting of all harmonic functions in B, where B is the ball in the n-dimensional Euclidean space and E is any Banach space. Consider also the space $h^{- \infty} (E *)$ consisting of all harmonic E*-valued functions g such that ${(1 - | x |)}^{m} f$ is bounded for some m>0. Then the dual $h^{\infty} (E *)$ is represented by $h^{- \infty} (E *)$ through $⟨ f, g ⟩_{0} = l i m_{r \to 1} ʃ_{B} ⟨ f (r x), g (x) ⟩ d x$ , $f \in h^{- \infty} (E *), g \in h^{\infty} (E)$ . This extends the results of S. Bell in the scalar case.

A class of functions containing polyharmonic functions in ℝⁿ

V. Anandam, M. Damlakhi (2003)

Annales Polonici Mathematici

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Some properties of the functions of the form $v (x) = \sum_{i = 0}^{m} {| x |}^{i} h_{i} (x)$ in ℝⁿ, n ≥ 2, where each $h_{i}$ is a harmonic function defined outside a compact set, are obtained using the harmonic measures.

On separately subharmonic functions (Lelong’s problem)

A. Sadullaev (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

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The main result of the present paper is : every separately-subharmonic function $u (x, y)$ , which is harmonic in $y$ , can be represented locally as a sum two functions, $u = u^{*} + U$ , where $U$ is subharmonic and $u^{*}$ is harmonic in $y$ , subharmonic in $x$ and harmonic in $(x, y)$ outside of some nowhere dense set $S$ .

Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents

Fanghua Lin, Tristan Rivière (1999)

Journal of the European Mathematical Society

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There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a $S^{1}$ -valued function defined on the boundary of a bounded regular domain of $R^{n}$ . When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within...

A critical growth rate for harmonic and subharmonic functions in the open ball in $R^{n}$

Krzysztof Samotij (1987)

Colloquium Mathematicae

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On the fusion problem for degenerate elliptic equations II

Stephen M. Buckley, Pekka Koskela (1999)

Commentationes Mathematicae Universitatis Carolinae

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Let $F$ be a relatively closed subset of a Euclidean domain $Ω$ . We investigate when solutions $u$ to certain elliptic equations on $Ω ∖ F$ are restrictions of solutions on all of $Ω$ . Specifically, we show that if $\partial F$ is not too large, and $u$ has a suitable decay rate near $F$ , then $u$ can be so extended.

Landau's theorem for p-harmonic mappings in several variables

Sh. Chen, S. Ponnusamy, X. Wang (2012)

Annales Polonici Mathematici

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A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation $Δ^{p} f = 0$ , where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function $f : Ω \to ℂ^{m}$ is said to be p-harmonic in Ω if each component function $f_{i}$ (i∈ 1,...,m) of $f = (f ₁, . . ., f_{m})$ is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings...

On the order of starlikeness and convexity of complex harmonic functions with a two-parameter coefficient condition

Agnieszka Sibelska (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The article of J. Clunie and T. Sheil-Small [3], published in 1984, intensified the investigations of complex functions harmonic in the unit disc $Δ$ . In particular, many papers about some classes of complex mappings with the coefficient conditions have been published. Consideration of this type was undertaken in the period 1998–2004 by Y. Avci and E. Złotkiewicz [2], A. Ganczar [5], Z. J. Jakubowski, G. Adamczyk, A. Łazinska and A. Sibelska [1], [8], [7], H. Silverman [12] and J. M. Jahangiri...

Hörmander systems and harmonic morphisms

Elisabetta Barletta (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Given a Hörmander system $X = {X_{1}, \dots, X_{m}}$ on a domain $Ω \subseteq 𝐑^{n}$ we show that any subelliptic harmonic morphism $φ$ from $Ω$ into a $ν$ -dimensional riemannian manifold $N$ is a (smooth) subelliptic harmonic map (in the sense of J. Jost & C-J. Xu, [9]). Also $φ$ is a submersion provided that $ν \leq m$ and $X$ has rank $m$ . If $Ω = 𝐇_{n}$ (the Heisenberg group) and $X = \{\frac{1}{2} (L_{α} + L_{\bar{α}}), \frac{1}{2 i} (L_{α} - L_{\bar{α}})\}$ , where $L_{\bar{α}} = \partial / \partial {\bar{z}}^{α} - i z^{α} \partial / \partial t$ is the Lewy operator, then a smooth map $φ : Ω \to N$ is a subelliptic harmonic morphism if and only if $φ \circ π : (C (𝐇_{n}), F_{θ_{0}}) \to N$ is a harmonic morphism, where $S^{1} \to C (𝐇_{n}) \overset{π}{\to} \to 𝐇_{n}$ is the canonical circle bundle and $F_{θ_{0}}$ ...

On Polynomially Bounded Harmonic Functions on the $Z^{d}$ Lattice

Piotr Nayar (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove that if $f : Z^{d} \to R$ is harmonic and there exists a polynomial $W : Z^{d} \to R$ such that f + W is nonnegative, then f is a polynomial.

Harmonic measures for symmetric stable processes

Jang-Mei Wu (2002)

Studia Mathematica

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Let D be an open set in ℝⁿ (n ≥ 2) and ω(·,D) be the harmonic measure on $D^{c}$ with respect to the symmetric α-stable process (0 < α < 2) killed upon leaving D. We study inequalities on volumes or capacities which imply that a set S on ∂D has zero harmonic measure and others which imply that S has positive harmonic measure. In general, it is the relative sizes of the sets S and $D^{c} ∖ S$ that determine whether ω(S,D) is zero or positive.

Uniform bounds for quotients of Green functions on $C^{1, 1}$ -domains

H. Hueber, M. Sieveking (1982)

Annales de l'institut Fourier

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Let $Δ u = Σ_{i} \frac{\partial^{2}}{\partial_{x_{i}}^{2}}$ , $L u = Σ_{i, j} a_{i j} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} u + Σ_{i} b_{i} \frac{\partial}{\partial x_{i}} u + c u$ be elliptic operators with Hölder continuous coefficients on a bounded domain $Ω \subset R^{n}$ of class $C^{1, 1}$ . There is a constant $c > 0$ depending only on the Hölder norms of the coefficients of $L$ and its constant of ellipticity such that $c^{- 1} G_{Δ}^{Ω} \leq G_{L}^{Ω} \leq c G_{Δ}^{Ω} on Ω \times Ω,$ where $γ_{Δ}^{Ω}$ (resp. $G_{L}^{Ω}$ ) are the Green functions of $Δ$ (resp. $L$ ) on $Ω$ .