A counterexample to the $L^{p}$-Hodge decomposition

Piotr Hajłasz

Displaying similar documents to “A counterexample to the $L^{p}$ -Hodge decomposition”

Finite distortion functions and Douglas-Dirichlet functionals

Qingtian Shi (2019)

Czechoslovak Mathematical Journal

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In this paper, we estimate the Douglas-Dirichlet functionals of harmonic mappings, namely Euclidean harmonic mapping and flat harmonic mapping, by using the extremal dilatation of finite distortion functions with given boundary value on the unit circle. In addition, $\bar{\partial}$ -Dirichlet functionals of harmonic mappings are also investigated.

Generalized Hölder type spaces of harmonic functions in the unit ball and half space

Alexey Karapetyants, Joel Esteban Restrepo (2020)

Czechoslovak Mathematical Journal

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We study spaces of Hölder type functions harmonic in the unit ball and half space with some smoothness conditions up to the boundary. The first type is the Hölder type space of harmonic functions with prescribed modulus of continuity $ω = ω (h)$ and the second is the variable exponent harmonic Hölder space with the continuity modulus ${| h |}^{λ (\cdot)}$ . We give a characterization of functions in these spaces in terms of the behavior of their derivatives near the boundary.

Dirichlet problem with L $^{p}$ -boundary data in contractible domains of Carnot groups

Andrea Bonfiglioli, Ermanno Lanconelli (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let $ℒ$ be a sub-laplacian on a stratified Lie group $G$ . In this paper we study the Dirichlet problem for $ℒ$ with $L^{p}$ -boundary data, on domains $Ω$ which are contractible with respect to the natural dilations of $G$ . One of the main difficulties we face is the presence of non-regular boundary points for the usual Dirichlet problem for $ℒ$ . A potential theory approach is followed. The main results are applied to study a suitable notion of Hardy spaces.

Carleson measures for weighted harmonic mixed norm spaces on bounded domains in $ℝ^{n}$

Ivana Savković (2022)

Czechoslovak Mathematical Journal

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We study weighted mixed norm spaces of harmonic functions defined on smoothly bounded domains in $ℝ^{n}$ . Our principal result is a characterization of Carleson measures for these spaces. First, we obtain an equivalence of norms on these spaces. Then we give a necessary and sufficient condition for the embedding of the weighted harmonic mixed norm space into the corresponding mixed norm space.

Behavior of biharmonic functions on Wiener's and Royden's compactifications

Y. K. Kwon, Leo Sario, Bertram Walsh (1971)

Annales de l'institut Fourier

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Let $R$ be a smooth Riemannian manifold of finite volume, $Δ$ its Laplace (-Beltrami) operator. Canonical direct-sum decompositions of certain subspaces of the Wiener and Royden algebras of $R$ are found, and for biharmonic functions (those for which $Δ Δ u = 0$ ) the decompositions are related to the values of the functions and their Laplacians on appropriate ideal boundaries.

$p$ Harmonic Measure in Simply Connected Domains

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

Annales de l’institut Fourier

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Let $Ω$ be a bounded simply connected domain in the complex plane, $ℂ$ . Let $N$ be a neighborhood of $\partial Ω$ , let $p$ be fixed, $1 < p < \infty,$ and let $\hat{u}$ be a positive weak solution to the $p$ Laplace equation in $Ω \cap N .$ Assume that $\hat{u}$ has zero boundary values on $\partial Ω$ in the Sobolev sense and extend $\hat{u}$ to $N ∖ Ω$ by putting $\hat{u} \equiv 0$ on $N ∖ Ω .$ Then there exists a positive finite Borel measure $\hat{μ}$ on $ℂ$ with support contained in $\partial Ω$ and such that $\begin{matrix} \int | \nabla \hat{u} |^{p - 2} 〈 \nabla \hat{u}, \nabla φ 〉 d A = - \int φ d \hat{μ} \end{matrix}$ whenever $φ \in C_{0}^{\infty} (N) .$ If $p = 2$ and if $\hat{u}$ is the Green function for $Ω$ with pole at $x \in Ω ∖ \bar{N}$ then the...

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 $Ω$ .

Displaying similar documents to “A counterexample to the L p -Hodge decomposition”

Finite distortion functions and Douglas-Dirichlet functionals

Generalized Hölder type spaces of harmonic functions in the unit ball and half space

Dirichlet problem with L p -boundary data in contractible domains of Carnot groups

Carleson measures for weighted harmonic mixed norm spaces on bounded domains in ℝ n

Behavior of biharmonic functions on Wiener's and Royden's compactifications

p Harmonic Measure in Simply Connected Domains

Uniform bounds for quotients of Green functions on C 1 , 1 -domains

Displaying similar documents to “A counterexample to the $L^{p}$ -Hodge decomposition”

Dirichlet problem with L $^{p}$ -boundary data in contractible domains of Carnot groups

Carleson measures for weighted harmonic mixed norm spaces on bounded domains in $ℝ^{n}$

$p$ Harmonic Measure in Simply Connected Domains

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