Hardy spaces for the Laplacian with lower order perturbations

Tomasz Luks

Displaying similar documents to “Hardy spaces for the Laplacian with lower order perturbations”

Properties of harmonic conjugates

Paweł Sobolewski (2008)

Annales UMCS, Mathematica

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We give a new proof of Hardy and Littlewood theorem concerning harmonic conjugates of functions u such that ∫D |u(z)|pdA(z) < ∞, 0 < p ≤ 1. We also obtain an inequality for integral means of such harmonic functions u.

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 $S$ .

Duality on vector-valued weighted harmonic Bergman spaces

Salvador Pérez-Esteva (1996)

Studia Mathematica

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We study the duals of the spaces $A^{p α} (X)$ of harmonic functions in the unit ball of $ℝ^{n}$ with values in a Banach space X, belonging to the Bochner $L^{p}$ space with weight ${(1 - | x |)}^{α}$ , denoted by $L^{p α} (X)$ . For 0 < α < p-1 we construct continuous projections onto $A^{p α} (X)$ providing a decomposition $L^{p α} (X) = A^{p α} (X) + M^{p α} (X)$ . We discuss the conditions on p, α and X for which $A^{p α} (X) * = A^{q α} (X *)$ and $M^{p α} (X) * = M^{q α} (X *)$ , 1/p+1/q = 1. The last equality is equivalent to the Radon-Nikodým property of X*.

Boundary behaviour of harmonic functions in a half-space and brownian motion

D. L. Burkholder, Richard F. Gundy (1973)

Annales de l'institut Fourier

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Let $u$ be harmonic in the half-space $R_{+}^{n + 1}$ , $n \geq 2$ . We show that $u$ can have a fine limit at almost every point of the unit cubs in $R^{n} = \partial R_{+}^{n + 1}$ but fail to have a nontangential limit at any point of the cube. The method is probabilistic and utilizes the equivalence between conditional Brownian motion limits and fine limits at the boundary. In $R_{+}^{2}$ it is known that the Hardy classes $H^{p}$ , $0 < p < \infty$ , may be described in terms of the integrability of the nontangential maximal function, or, alternatively, in terms...

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 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.

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.

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...

Biharmonic morphisms

Mustapha Chadli, Mohamed El Kadiri, Sabah Haddad (2005)

Commentationes Mathematicae Universitatis Carolinae

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Let $(X, ℋ)$ and $(X^{'}, ℋ^{'})$ be two strong biharmonic spaces in the sense of Smyrnelis whose associated harmonic spaces are Brelot spaces. A biharmonic morphism from $(X, ℋ)$ to $(X^{'}, ℋ^{'})$ is a continuous map from $X$ to $X^{'}$ which preserves the biharmonic structures of $X$ and $X^{'}$ . In the present work we study this notion and characterize in some cases the biharmonic morphisms between $X$ and $X^{'}$ in terms of harmonic morphisms between the harmonic spaces associated with $(X, ℋ)$ and $(X^{'}, ℋ^{'})$ and the coupling kernels of them.