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

Ivana Savković

Displaying similar documents to “Carleson measures for weighted harmonic mixed norm spaces on bounded domains in $ℝ^{n}$ ”

Inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball

Ömer Faruk Doğan, Adem Ersin Üreyen (2019)

Czechoslovak Mathematical Journal

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We consider harmonic Bergman-Besov spaces $b_{α}^{p}$ and weighted Bloch spaces $b_{α}^{\infty}$ on the unit ball of $ℝ^{n}$ for the full ranges of parameters $0 < p < \infty$ , $α \in ℝ$ , and determine the precise inclusion relations among them. To verify these relations we use Carleson measures and suitable radial differential operators. For harmonic Bergman spaces various characterizations of Carleson measures are known. For weighted Bloch spaces we provide a characterization when $α > 0$ .

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.

Some $R^{N}$ capacities and applications - the lemma on mixed derivatives revisited

J. Korevaar (1986)

Matematički Vesnik

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The diagonal mapping in mixed norm spaces

Guangbin Ren, Jihuai Shi (2004)

Studia Mathematica

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For any holomorphic function F in the unit polydisc Uⁿ of ℂⁿ, we consider its restriction to the diagonal, i.e., the function in the unit disc U of ℂ defined by F(z) = F(z,...,z), and prove that the diagonal mapping maps the mixed norm space $H^{p, q, α} (U ⁿ)$ of the polydisc onto the mixed norm space $H^{p, q, | α | + (p / q + 1) (n - 1)} (U)$ of the unit disc for any 0 < p < ∞ and 0 < q ≤ ∞.

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 existence of exactly one solution of integral equations in the space $L^{P}$ with a mixed norm

Bogdan Rzepecki (1975)

Annales Polonici Mathematici

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An example of the equation $u_{t} = u_{x x} + f (x, t, u)$ with distinct maximum and minimum solutions of a mixed problem

W. Mlak (1963)

Annales Polonici Mathematici

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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.

$L^{p}$ harmonic $1$ -form on submanifold with weighted Poincaré inequality

Xiaoli Chao, Yusha Lv (2018)

Czechoslovak Mathematical Journal

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We deal with complete submanifolds with weighted Poincaré inequality. By assuming the submanifold is $δ$ -stable or has sufficiently small total curvature, we establish two vanishing theorems for $L^{p}$ harmonic $1$ -forms, which are extensions of the results of Dung-Seo and Cavalcante-Mirandola-Vitório.

Some characterizations of harmonic Bloch and Besov spaces

Xi Fu, Bowen Lu (2016)

Czechoslovak Mathematical Journal

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The relationship between weighted Lipschitz functions and analytic Bloch spaces has attracted much attention. In this paper, we define harmonic $ω$ - $α$ -Bloch space and characterize it in terms of $ω ((1 - {| x |}^{2})^{β} {(1 - | y |}^{2})^{α - β}) | \frac{f (x) - f (y)}{x - y} |$ and $ω ((1 - {| x |}^{2})^{β} {(1 - | y |}^{2})^{α - β}) | \frac{f (x) - f (y)}{| x | y - x^{'}} |$ where $ω$ is a majorant. Similar results are extended to harmonic little $ω$ - $α$ -Bloch and Besov spaces. Our results are generalizations of the corresponding ones in G. Ren, U. Kähler (2005).

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.

Variations of mixed Hodge structure attached to the deformation theory of a complex variation of Hodge structures

Philippe Eyssidieux, Carlos Simpson (2011)

Journal of the European Mathematical Society

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Let $X$ be a compact Kähler manifold, $x \in X$ be a base point and $ρ : π_{1} (X, x) \to G L_{N} (C)$ be the monodromy representation of a $𝒞$ -VHS. Building on Goldman–Millson’s classical work, we construct a mixed Hodge structure on the complete local ring of the representation variety at $ρ$ and a variation of mixed Hodge structures whose monodromy is the universal deformation of $ρ$ .

Mixed $A_{p} - A_{\infty}$ estimates with one supremum

Andrei K. Lerner, Kabe Moen (2013)

Studia Mathematica

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We establish several mixed $A_{p} - A_{\infty}$ bounds for Calderón-Zygmund operators that only involve one supremum. We address both cases when the $A_{\infty}$ part of the constant is measured using the exponential-logarithmic definition and using the Fujii-Wilson definition. In particular, we answer a question of the first author and provide an answer, up to a logarithmic factor, to a conjecture of Hytönen and Lacey. Moreover, we give an example to show that our bounds with the logarithmic factors can be arbitrarily...

ℓ¹-Spreading models in subspaces of mixed Tsirelson spaces

Denny H. Leung, Wee-Kee Tang (2006)

Studia Mathematica

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We investigate the existence of higher order ℓ¹-spreading models in subspaces of mixed Tsirelson spaces. For instance, we show that the following conditions are equivalent for the mixed Tsirelson space $X = T [{(θ ₙ, ₙ)}_{n = 1}^{\infty}]$ : (1) Every block subspace of X contains an $ℓ ¹ -_{ω}$ -spreading model, (2) The Bourgain ℓ¹-index $I_{b} (Y) = I (Y) > ω^{ω}$ for any block subspace Y of X, (3) $l i m ₘ l i m s u p ₙ θ_{m + n} / θ ₙ > 0$ and every block subspace Y of X contains a block sequence equivalent to a subsequence of the unit vector basis of X. Moreover, if one (and hence all) of these...

On the isomorphism classes of weighted spaces of harmonic and holomorphic functions

Wolfgang Lusky (2006)

Studia Mathematica

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Let Ω be either the complex plane or the open unit disc. We completely determine the isomorphism classes of $H v = f : Ω \to ℂ h o l o m o r p h i c : s u p_{z \in Ω} | f (z) | v (z) < \infty$ and investigate some isomorphism classes of $h v = f : Ω \to ℂ h a r m o n i c : s u p_{z \in Ω} | f (z) | v (z) < \infty$ where v is a given radial weight function. Our main results show that, without any further condition on v, there are only two possibilities for Hv, namely either $H v \sim l_{\infty}$ or $H v \sim H_{\infty}$ , and at least two possibilities for hv, again $h v \sim l_{\infty}$ and $h v \sim H_{\infty}$ . We also discuss many new examples of weights.

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*.

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 $Ω$ .

On a question of T. Sheil-Small regarding valency of harmonic maps

Daoud Bshouty, Abdallah Lyzzaik (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The aim of this work is to answer positively a more general question than the following which is due to T. Sheil-Small: Does the harmonic extension in the open unit disc of a mapping f from the unit circle into itself of the form $f (e^{i t}) = e^{i φ (t)}$ , $0 \leq t \leq 2 π$ where $φ$ is a continuously non-decreasing function that satisfies $φ (2 π) - φ (0) = 2 N π$ , assume every value finitely many times in the disc?

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 Weighted Eigenvalue Problems Driven by both $p (\cdot)$ -Harmonic and $p (\cdot)$ -Biharmonic Operators

Mohamed Laghzal, Abdelouahed El Khalil, Abdelfattah Touzani (2021)

Communications in Mathematics

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The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both $p (\cdot)$ -Harmonic and $p (\cdot)$ -biharmonic operators $\begin{matrix} Δ_{p (x)}^{2} u - Δ_{p (x)} u = λ w (x) {| u |}^{q (x) - 2} u in Ω, \\ u \in W^{2, p (\cdot)} (Ω) \cap W_{0}^{1, p (\cdot)} (Ω), \end{matrix}$ is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces $L^{p (\cdot)} (Ω)$ and $W^{m, p (\cdot)} (Ω)$ .

Harmonic functions in four variables with rational and algebraic $p_{4}$ -associates

R. P. Gilbert (1964)

Annales Polonici Mathematici

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Harmonie reflections

Lieven Vanhecke, Maria-Elena Vazquez-Abal (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We study local reflections $\phi_{\sigma}$ with respect to a curve $\sigma$ in a Riemannian manifold and prove that $\sigma$ is a geodesic if $\phi_{\sigma}$ is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if $\phi_{\sigma}$ is harmonic for all geodesies $\sigma$ .

Displaying similar documents to “Carleson measures for weighted harmonic mixed norm spaces on bounded domains in ℝ n ”

Displaying similar documents to “Carleson measures for weighted harmonic mixed norm spaces on bounded domains in $ℝ^{n}$ ”