Displaying similar documents to “Landau's theorem for p-harmonic mappings in several variables”

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 t 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?

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 ) = 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 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 ) | f ( x ) - f ( y ) x - y | and ω ( ( 1 - | x | 2 ) β ( 1 - | y | 2 ) α - β ) | 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).

Harmonie reflections

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

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

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

On the dimension of p -harmonic measure in space

John L. Lewis, Kaj Nyström, Andrew Vogel (2013)

Journal of the European Mathematical Society

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Let Ω n , n 3 , and let p , 1 < p < , p 2 , be given. In this paper we study the dimension of p -harmonic measures that arise from non-negative solutions to the p -Laplace equation, vanishing on a portion of Ω , in the setting of δ -Reifenberg flat domains. We prove, for p n , that there exists δ ˜ = δ ˜ ( p , n ) > 0 small such that if Ω is a δ -Reifenberg flat domain with δ < δ ˜ , then p -harmonic measure is concentrated on a set of σ -finite H n 1 -measure. We prove, for p n , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of p -harmonic...

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 ϕ σ with respect to a curve σ in a Riemannian manifold and prove that σ is a geodesic if ϕ σ is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if ϕ σ is harmonic for all geodesies σ .

On the integral representation of finely superharmonic functions

Abderrahim Aslimani, Imad El Ghazi, Mohamed El Kadiri (2019)

Commentationes Mathematicae Universitatis Carolinae

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In the present paper we study the integral representation of nonnegative finely superharmonic functions in a fine domain subset U of a Brelot 𝒫 -harmonic space Ω with countable base of open subsets and satisfying the axiom D . When Ω satisfies the hypothesis of uniqueness, we define the Martin boundary of U and the Martin kernel K and we obtain the integral representation of invariant functions by using the kernel K . As an application of the integral representation we extend to the cone...

Arithmetic theory of harmonic numbers (II)

Zhi-Wei Sun, Li-Lu Zhao (2013)

Colloquium Mathematicae

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For k = 1,2,... let H k denote the harmonic number j = 1 k 1 / j . In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have k = 1 p - 1 ( H k ) / ( k 2 k ) 7 / 24 p B p - 3 ( m o d p ² ) , k = 1 p - 1 ( H k , 2 ) / ( k 2 k ) - 3 / 8 B p - 3 ( m o d p ) , and k = 1 p - 1 ( H ² k , 2 n ) / ( k 2 n ) ( 6 n + 1 2 n - 1 + n ) / ( 6 n + 1 ) p B p - 1 - 6 n ( m o d p ² ) for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and H k , m : = j = 1 k 1 / ( j m ) .

Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales

Adam Osękowski (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let ( h k ) k 0 be the Haar system on [0,1]. We show that for any vectors a k from a separable Hilbert space and any ε k [ - 1 , 1 ] , k = 0,1,2,..., we have the sharp inequality | | k = 0 n ε k a k h k | | W ( [ 0 , 1 ] ) 2 | | k = 0 n a k h k | | L ( [ 0 , 1 ] ) , n = 0,1,2,..., where W([0,1]) is the weak- L space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound | | Y | | W ( Ω ) 2 | | X | | L ( Ω ) , where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.

A Weighted Eigenvalue Problems Driven by both p ( · ) -Harmonic and p ( · ) -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 ( · ) -Harmonic and p ( · ) -biharmonic operators Δ p ( x ) 2 u - Δ p ( x ) u = λ w ( x ) | u | q ( x ) - 2 u in Ω , u W 2 , p ( · ) ( Ω ) W 0 1 , p ( · ) ( Ω ) , is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces L p ( · ) ( Ω ) and W m , p ( · ) ( Ω ) .

Deformations of Metrics and Biharmonic Maps

Aicha Benkartab, Ahmed Mohammed Cherif (2020)

Communications in Mathematics

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We construct biharmonic non-harmonic maps between Riemannian manifolds ( M , g ) and ( N , h ) by first making the ansatz that ϕ : ( M , g ) ( N , h ) be a harmonic map and then deforming the metric on N by h ˜ α = α h + ( 1 - α ) d f d f to render ϕ biharmonic, where f is a smooth function with gradient of constant norm on ( N , h ) and α ( 0 , 1 ) . We construct new examples of biharmonic non-harmonic maps, and we characterize the biharmonicity of some curves on Riemannian manifolds.

Injectivity of sections of convex harmonic mappings and convolution theorems

Liulan Li, Saminathan Ponnusamy (2016)

Czechoslovak Mathematical Journal

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We consider the class 0 of sense-preserving harmonic functions f = h + g ¯ defined in the unit disk | z | < 1 and normalized so that h ( 0 ) = 0 = h ' ( 0 ) - 1 and g ( 0 ) = 0 = g ' ( 0 ) , where h and g are analytic in the unit disk. In the first part of the article we present two classes 𝒫 H 0 ( α ) and 𝒢 H 0 ( β ) of functions from 0 and show that if f 𝒫 H 0 ( α ) and F 𝒢 H 0 ( β ) , then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters α and β are satisfied. In the second part we study the harmonic sections...

The harmonic Cesáro and Copson operators on the spaces L p ( ) , 1 ≤ p ≤ 2

Ferenc Móricz (2002)

Studia Mathematica

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The harmonic Cesàro operator is defined for a function f in L p ( ) for some 1 ≤ p < ∞ by setting ( f ) ( x ) : = x ( f ( u ) / u ) d u for x > 0 and ( f ) ( x ) : = - - x ( f ( u ) / u ) d u for x < 0; the harmonic Copson operator ℂ* is defined for a function f in L ¹ l o c ( ) by setting * ( f ) ( x ) : = ( 1 / x ) x f ( u ) d u for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If f L p ( ) for some 1 ≤ p ≤ 2, then ( ( f ) ) ( t ) = * ( f ̂ ) ( t ) a.e., where f̂ denotes the Fourier transform of f. (ii) If f L p ( ) for some 1 < p ≤ 2, then...

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.

Interpolation theorem for the p-harmonic transform

Luigi D&amp;#039;Onofrio, Tadeusz Iwaniec (2003)

Studia Mathematica

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We establish an interpolation theorem for a class of nonlinear operators in the Lebesgue spaces s ( ) arising naturally in the study of elliptic PDEs. The prototype of those PDEs is the second order p-harmonic equation d i v | u | p - 2 u = d i v . In this example the p-harmonic transform is essentially inverse to d i v ( | | p - 2 ) . To every vector field q ( , ) our operator p assigns the gradient of the solution, p = u p ( , ) . The core of the matter is that we go beyond the natural domain of definition of this operator. Because of nonlinearity our...

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 h o l o m o r p h i c : s u p z Ω | f ( z ) | v ( z ) < and investigate some isomorphism classes of h v = f : Ω h a r m o n i c : s u p z Ω | f ( z ) | v ( z ) < 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 l or H v H , and at least two possibilities for hv, again h v l and h v H . We also discuss many new examples of weights.

Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity

Erwann Aubry, Colin Guillarmou (2011)

Journal of the European Mathematical Society

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For odd-dimensional Poincaré–Einstein manifolds ( X n + 1 , g ) , we study the set of harmonic k -forms (for k < n / 2 ) which are C m (with m ) on the conformal compactification X ¯ of X . This set is infinite-dimensional for small m but it becomes finite-dimensional if m is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology H k ( X ¯ , X ¯ ) and the kernel of the Branson–Gover [3] differential operators ( L k , G k ) on the conformal infinity ( X ¯ , [ h 0 ] ) . We also relate the set of C n - 2 k + 1 ( Λ k ( X ¯ ) ) forms in the kernel of d + δ g ...

Liouville theorems for self-similar solutions of heat flows

Jiayu Li, Meng Wang (2009)

Journal of the European Mathematical Society

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Let N be a compact Riemannian manifold. A quasi-harmonic sphere on N is a harmonic map from ( m , e | x | 2 / 2 ( m - 2 ) / d s 0 2 ) to N ( m 3 ) with finite energy ([LnW]). Here d s 2 0 is the Euclidean metric in m . Such maps arise from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target N . We also derive gradient estimates and Liouville theorems...

On a result by Clunie and Sheil-Small

Dariusz Partyka, Ken-ichi Sakan (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping F in the unit disk 𝔻 , if F ( 𝔻 ) is a convex domain, then the inequality | G ( z 2 ) - G ( z 1 ) | < | H ( z 2 ) - H ( z 1 ) | holds for all distinct points z 1 , z 2 𝔻 . Here H and G are holomorphic mappings in 𝔻 determined by F = H + G ¯ , up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain Ω in and improve it provided F is additionally a quasiconformal mapping...

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 on the unit ball of n for the full ranges of parameters 0 < p < , α , 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 .

Natural pseudodistances between closed surfaces

Pietro Donatini, Patrizio Frosini (2007)

Journal of the European Mathematical Society

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Let us consider two closed surfaces , 𝒩 of class C 1 and two functions ϕ : , ψ : 𝒩 of class C 1 , called measuring functions. The natural pseudodistance d between the pairs ( , ) , ( 𝒩 , ψ ) is defined as the infimum of Θ ( f ) : = max P | ϕ ( P ) ψ ( f ( P ) ) | as f varies in the set of all homeomorphisms from onto 𝒩 . In this paper we prove that the natural pseudodistance equals either | c 1 c 2 | , 1 2 | c 1 c 2 | , or 1 3 | c 1 c 2 | , where c 1 and c 2 are two suitable critical values of the measuring functions. This shows that a previous relation between the natural pseudodistance and...

A characterization of the Riemann extension in terms of harmonicity

Cornelia-Livia Bejan, Şemsi Eken (2017)

Czechoslovak Mathematical Journal

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If ( M , ) is a manifold with a symmetric linear connection, then T * M can be endowed with the natural Riemann extension g ¯ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to g ¯ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure 𝒫 on ( T * M , g ¯ ) and prove that 𝒫 is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if g ¯ reduces...