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The total Q-curvature, volume entropy and polynomial growth polyharmonic functions

ADVANCES IN MATHEMATICS 2024年 450卷

作者： Li, Mingxiang Nanjing Univ Nanjing Peoples R China

In this paper, we investigate a conformally flat and complete manifold (M, g ) = ( R n , e 2u |dx | 2 ) with finite total Qcurvature. We introduce a new volume entropy, incorporating the background Euclidean metric, and demonstrate that the metric g is normal if and only if the volume entropy is finite. Furthermore, we establish an identity for the volume entropy utilizing the integrated Q -curvature. Additionally, under normal metric assumption, we get a result concerning the behavior of the geometric distance at infinity compared with Euclidean distance. With help of this result, we prove that each polynomial growth polyharmonic function on such manifolds is of finite dimension. Meanwhile, we prove several rigidity results by imposing restrictions on the sign of the Qcurvature. Specifically, we establish that on such manifolds, the Cohn-Vossen inequality achieves equality if and only if each polynomial growth polyharmonic function is a constant. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Total Q-curvature Normal solution Volume entropy polyharmonic functions

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WEIGHTED INTEGRABILITY OF polyharmonic functions IN THE HIGHER-DIMENSIONAL CASE

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ANALYSIS & PDE 2021年第7期14卷 2047-2068页

作者： Liu, Congwen Perala, Antti Si, Jiajia Univ Sci & Technol China Sch Math Sci CAS Wu Wen Tsun Key Lab Math Hefei Peoples R China Umea Univ Dept Math & Math Stat Umea Sweden Hainan Univ Sch Sci Haikou Hainan Peoples R China

This paper is concerned with the L-p integrability of N-harmonic functions with respect to the standard weights (1 - vertical bar x vertical bar(2))(alpha) on the unit ball B of R-n, n >= 2. More precisely, our goal is to determine the real (negative) parameters alpha for which (1 - vertical bar x vertical bar(2))(alpha/p) u(x) is an element of L-p(B) implies that u equivalent to 0 whenever u is a solution of the N-Laplace equation on B. This question is motivated by the uniqueness considerations of the Dirichlet problem for the N-Laplacian Delta(N). Our study is inspired by a recent work of Borichev and Hedenmalm (Adv. Math. 264 (2014), 464-505), where a complete answer to the above question in the case n D 2 is given for the full scale 0 < p < infinity. When n >= 3, we obtain an analogous characterization for n-2/n-1 <= p < infinity and remark that the remaining case can be genuinely more difficult. Also, we extend the remarkable cellular decomposition theorem of Borichev and Hedenmalm to all dimensions.

关键词： polyharmonic functions weighted integrability boundary behavior cellular decomposition

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Boundary Representations of λ-Harmonic and polyharmonic functions on Trees

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POTENTIAL ANALYSIS 2019年第4期51卷 541-561页

作者： Picardello, Massimo A. Woess, Wolfgang Univ Roma Tor Vergata Dipartimento Matemat I-00133 Rome Italy Graz Univ Technol Inst Diskrete Math Steyrergasse 30 A-8010 Graz Austria

On a countable tree T, allowing vertices with infinite degree, we consider an arbitrary stochastic irreducible nearest neighbour transition operator P. We provide a boundary integral representation for general eigenfunctions of P with eigenvalue lambda is an element of C. This is possible whenever lambda is in the resolvent set of P as a self-adjoint operator on a suitable l(2)-space and the diagonal elements of the resolvent ("Green function") do not vanish at lambda. We show that when P is invariant under a transitive (not necessarily fixed-point-free) group action, the latter condition holds for all lambda not equal 0 in the resolvent set. These results extend and complete previous results by Cartier, by Figa-Talamanca and Steger, and by Woess. For those eigenvalues, we also provide an integral representation of lambda-polyharmonic functions of any order n, that is, functions f:T -> C for which (lambda I - P)(n)f = 0. This is a far-reaching extension of work of Cohen et al., who provided such a representation for the simple random walk on a homogeneous tree and eigenvalue lambda = 1. Finally, we explain the (much simpler) analogous results for "forward only" transition operators, sometimes also called martingales on trees.

关键词： Tree Stochastic transition operator lambda-harmonic functions polyharmonic functions Martin kernel Boundary integral

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Spherical polyharmonics and Poisson kernels for polyharmonic functions

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COMPLEX VARIABLES AND ELLIPTIC EQUATIONS 2019年第3期64卷 420-442页

作者： Grzebula, Hubert Michalik, Slawomir Cardinal Stefan Wyszynski Univ Coll Sci Fac Math & Nat Sci Warsaw Poland

We introduce and develop the notion of spherical polyharmonics, which are a natural generalisation of spherical harmonics. In particular we study the theory of zonal polyharmonics, which allows us, analogously to zonal harmonics, to construct Poisson kernels for polyharmonic functions on the union of rotated balls. We find the representation of Poisson kernels and zonal polyharmonics in terms of the Gegenbauer polynomials. We show the connection between the classical Poisson kernel for harmonic functions on the ball, Poisson kernels for polyharmonic functions on the union of rotated balls, and the Cauchy-Hua kernel for holomorphic functions on the Lie ball.

关键词： Spherical polyharmonics zonal polyharmonics polyharmonic functions Poisson kernel Gegenbauer polynomials Cauchy-Hua kernel

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A symmetry property for polyharmonic functions vanishing on equidistant hyperplanes

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MATHEMATISCHE NACHRICHTEN 2017年第7期290卷 1087-1096页

作者： Kounchev, Ognyan Render, Hermann Bulgarian Acad Sci Inst Math & Informat Acad G Bonchev St 8 BU-1113 Sofia Bulgaria IZKS Univ Bonn Bonn Germany Univ Coll Dublin Sch Math & Stat Dublin 4 Ireland

Let u (t, y) be a polyharmonic function of order N defined on the strip (a, b) x R-d satisfying the growth condition sup(t is an element of k)vertical bar u(t, y)vertical bar infinity and any compact subinterval K of... 详细信息

Let u (t, y) be a polyharmonic function of order N defined on the strip (a, b) x R-d satisfying the growth condition sup(t is an element of k)vertical bar u(t, y)vertical bar <= 0 (vertical bar y vertical bar((1-d)/2) e(pi/c vertical bar y vertical bar)) for vertical bar y vertical bar -> infinity and any compact subinterval K of (a, b), and suppose that u (t, y) vanishes on 2N - 1 equidistant hyperplanes of the form {t(j)} x R-d for t(j) = t(0) + jc is an element of (a, b) and j = - (N - 1), ..., N - 1. Then it is shown that u (t, y) is odd at t(0), i.e. that u (t(0) + t, y) = - u (t(0) -t, y) for y is an element of R-d. The second main result states that u is identically zero provided that u satisfies (0.1) and vanishes on 2N equidistant hyperplanes with distance c. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

关键词： polyharmonic functions polyanalytic functions multivariate interpolation polyharmonic interpolation

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A generalized mean value property for polyharmonic functions

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NUMERICAL ALGORITHMS 2016年第1期73卷 157-165页

作者： Floater, Michael S. Univ Oslo Dept Math Moltke Moes Vei 35 N-0851 Oslo Norway

A well known property of a harmonic function in a ball is that its value at the centre equals the mean of its values on the boundary. Less well known is the more general property that its value at any point x equals the mean over all chords through x of its values at the ends of the chord, linearly interpolated at x. In this paper we show that a similar property holds for polyharmonic functions of any order when linear interpolation is replaced by two-point Hermite interpolation of odd degree.

关键词： polyharmonic functions Polynomial interpolation Integral formulas

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Weighted integrability of polyharmonic functions

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ADVANCES IN MATHEMATICS 2014年 264卷 464-505页

作者： Borichev, Alexander Hedenmalm, Haakan Aix Marseille Univ CMI Lab Anal F-13453 Marseille 13 France KTH Royal Inst Technol Dept Math S-10044 Stockholm Sweden

To address the uniqueness issues associated with the Dirichlet problem for the N-harmonic equation on the unit disk D in the plane, we investigate the L-P integrability of N-harmonic functions with respect to the standard weights (1 vertical bar z vertical bar(2))(alpha). The question at hand is the following. If u solves Delta(N)u = 0 in D, where Delta stands for the Laplacian, and integral(D)vertical bar u(Z)vertical bar(p)(1 - vertical bar z vertical bar(2))(alpha)dA(z) < +infinity, must then u(z) 0? Here, N is a positive integer, alpha is real, and 0 < p < +infinity;dA is the usual area element. The answer will, generally speaking, depend on the triple (N, p, alpha). The most interesting case is 0 < p < 1. For a given N, we find an explicit critical curve p bar right arrow beta(N, p) - a piecewise affine function - such that for alpha > beta(N, p) there exist nontrivial functions u with Delta Nu = 0 of the given integrability, while for alpha <= beta(N, p), only u(z) 0 is possible. We also investigate the obstruction to uniqueness for the Dirichlet problem, that is, we study the structure of the functions in PHN, alpha p (D) when this space is nontrivial. We find a new structural decomposition of the polyharmonic functions - the cellular decomposition - which decomposes the polyharmonic weighted LP space in a canonical fashion. Corresponding to the cellular expansion is a tiling of part of the (p, alpha) plane into cells. The above uniqueness for the Dirichlet problem may be considered for any elliptic operator of order 2N. However, the above-mentioned critical integrability curve will depend rather strongly on the given elliptic operator, even in the constant coefficient case, for N > 1. Published by Elsevier Inc.

关键词： polyharmonic functions Weighted integrability Boundary behavior Cellular decomposition

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Spherical means of super-polyharmonic functions in the unit ball

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HIROSHIMA MATHEMATICAL JOURNAL 2014年第2期44卷 235-245页

作者： Futamura, Toshihide Mizuta, Yoshihiro Ohno, Takao Daido Univ Dept Math Nagoya Aichi 4578530 Japan Hiroshima Inst Technol Dept Mech Syst Engn Saeki Ku Hiroshima 7315193 Japan Oita Univ Fac Educ & Welf Sci Oita 8701192 Japan

For a super-polyharmonic function u on the unit ball satisfying a growth condition on spherical means, we study a growth property of the Riesz measure of u near the boundary.

关键词： polyharmonic functions super-polyharmonic functions spherical means Riesz decomposition Riesz measure

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polyharmonic potential theory on the Poincaré disk

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JOURNAL OF FUNCTIONAL ANALYSIS 2024年第9期286卷

作者： Picardello, Massimo A. Salvatori, Maura Woess, Wolfgang Univ Roma Tor Vergata Dipartimento Matemat I-00133 Rome Italy Univ Milan Dipartimento Matemat Via Saldini 50 I-20133 Milan Italy Graz Univ Technol Inst Diskrete Math Steyrergasse 30 A-8010 Graz Austria

We consider the open unit disk D equipped with the hyperbolic metric and the associated hyperbolic Laplacian 2. For lambda is an element of C and n is an element of N, a lambda-polyharmonic function of order n is a function f : D -> C such that (2 - lambda I)nf = 0. If n = 1, one gets lambda-harmonic functions. Based on a Theorem of Helgason on the latter functions, we prove a boundary integral representation theorem for lambda-polyharmonic functions. For this purpose, we first determine nth-order lambda-Poisson kernels. Subsequently, we introduce the lambda-polyspherical functions and determine their asymptotics at the boundary partial differential D, i.e., the unit circle. In particular, this proves that, for eigenvalues not in the interior of the L2-spectrum, the zeroes of these functions do not accumulate at the boundary circle. Hence the polyspherical functions can be used to normalise the nth- order Poisson kernels. By this tool, we extend to this setting several classical results of potential theory: namely, we study the boundary behaviour of lambda-polyharmonic functions, starting with Dirichlet and Riquier type problems and then proceeding to Fatou type admissible boundary limits. (c) 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// *** /licenses /by /4 .0/).

关键词： Hyperbolic Laplacian polyharmonic functions Analytic functionals Riquier and Fatou problems

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The universal approximation theorem for complex-valued neural networks

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APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 2023年第1期64卷 33-61页

作者： Voigtlaender, Felix Tech Univ Munich Dept Math D-85748 Garching Germany Univ Vienna Fac Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria Catholic Univ Eichstatt Ingolstadt KU Math Inst Machine Learning & Data Sci MIDS Schanz 49 D-85049 Ingolstadt Germany

We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a complex activation function Sigma : C-+ C in which each neuron per -forms the operation CN-+ C, z-+ Sigma(b +wT z) with weights w E CN and a bias b E C. We completely characterize those activation functions Sigma for which the asso-ciated complex networks have the universal approximation property, meaning that they can uniformly approximate any continuous function on any compact subset of Cd arbitrarily well. Unlike the classical case of real networks, the set of "good activation functions"-which give rise to networks with the universal approximation property-differs significantly depending on whether one considers deep networks or shallow networks: For deep networks with at least two hidden layers, the universal approximation property holds as long as Sigma is neither a polynomial, a holomorphic function, nor an antiholomorphic function. Shallow networks, on the other hand, are universal if and only if the real part or the imaginary part of Sigma is not a polyharmonic function.(c) 2022 Elsevier Inc. All rights reserved.

关键词： Complex-valued neural networks Universal approximation theorem Deep neural networks polyharmonic functions Holomorphic functions

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