scholarly journals Banach spaces of bounded solutions of Δu = Pu (P ≥ 0) on hyperbolic riemann surfaces

1974 ◽  
Vol 53 ◽  
pp. 141-155 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Banach Space ◽  
Riemann Surface ◽  
Banach Spaces ◽  
Riemann Surfaces ◽  
Space Structure ◽  
Bounded Solutions ◽  
Linear Isometry ◽  
Hölder Continuous ◽  
Hyperbolic Riemann Surfaces ◽  
Hyperbolic Riemann Surface

Consider a nonnegative Hölder continuous 2-form P(z)dxdy on a hyperbolic Riemann surface R (z = x + iy). We denote by PB(R) the Banach space of solutions of the equation Δu = Pu on R with finite supremum norms. We are interested in the question how the Banach space structure of PB(R) depends on P. Precisely we consider two such 2-forms P and Q on R and compare PB(R) and QB(R). If there exists a bijective linear isometry T of PB(R) to QB(R), then we say that PB(R) and QB(R) are isomorphic.

Qp Spaces on Riemann Surfaces

1998 ◽  
Vol 50 (3) ◽  
pp. 449-464 ◽  
Author(s):  
Rauno Aulaskari ◽  
Yuzan He ◽  
Juha Ristioja ◽  
Ruhan Zhao
Keyword(s):  
Banach Space ◽  
Riemann Surface ◽  
Unit Disk ◽  
Complex Plane ◽  
Riemann Surfaces ◽  
Bloch Space ◽  
Dirichlet Space ◽  
Function Spaces ◽  
Hyperbolic Riemann Surfaces ◽  
Qp Spaces

AbstractWe study the function spaces Qp(R) defined on a Riemann surface R, which were earlier introduced in the unit disk of the complex plane. The nesting property Qp(R) ⊆Qq(R) for 0 < p < q < ∞ is shown in case of arbitrary hyperbolic Riemann surfaces. Further, it is proved that the classical Dirichlet space AD(R) ⊆ Qp(R) for any p, 0 < p < ∞, thus sharpening T. Metzger's well-known result AD(R) ⊆ BMOA(R). Also the first author's result AD(R) ⊆ VMOA(R) for a regular Riemann surface R is sharpened by showing that, in fact, AD(R) ⊆ Qp,0(R) for all p, 0 < p < ∞. The relationships between Qp(R) and various generalizations of the Bloch space on R are considered. Finally we show that Qp(R) is a Banach space for 0 < p < ∞.

scholarly journals Entropy for hyperbolic Riemann surface laminations I

2014 ◽  
Author(s):  
Tien-Cuong Dinh ◽  
Viet-Anh Nguyen ◽  
Nessim Sibony
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Careful Analysis ◽  
Probability Measures ◽  
Metric Entropy ◽  
Hölder Continuous ◽  
Poincaré Metric ◽  
Singular Foliations ◽  
Open Questions ◽  
Hyperbolic Riemann Surface

This chapter introduces a notion of entropy for possibly singular hyperbolic laminations by Riemann surfaces. It also studies the transverse regularity of the Poincaré metric and the finiteness of the entropy. The chapter first focuses on compact laminations, which are transversally smooth, before turning to the case of singular foliations, showing how the Poincaré metric on leaves is transversally Hölder continuous. In addition, the chapter considers the problem in the proof that the entropy is finite for singular foliations is quite delicate and requires a careful analysis of the dynamics around the singularities. Finally, the chapter discusses a notion of metric entropy for harmonic probability measures and gives some open questions.

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

scholarly journals Analytic Functions on Some Riemann Surfaces

1963 ◽  
Vol 22 ◽  
pp. 211-217 ◽  
Author(s):  
Nobushige Toda ◽  
Kikuji Matsumoto
Keyword(s):  
Riemann Surface ◽  
Compact Subset ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Open Riemann Surface ◽  
Interesting Theorem ◽  
Hyperbolic Riemann Surface

Some years ago, Kuramochi gave in his paper [5] a very interesting theorem, which can be stated as follows.THEOREM OF KURAMOCHI. Let R be a hyperbolic Riemann surface of the class Of OHR(OHD,resp.). Then, for any compact subset K of R such that R—K is connected, R—K as an open Riemann surface belongs to the class 0AB(OAD resp.).

scholarly journals Limits of canonical forms on towers of Riemann surfaces

2020 ◽  
Vol 2020 (764) ◽  
pp. 287-304
Author(s):  
Hyungryul Baik ◽  
Farbod Shokrieh ◽  
Chenxi Wu
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Type Theorem ◽  
Universal Cover ◽  
Canonical Forms ◽  
Classical Version ◽  
Galois Cover ◽  
Hyperbolic Form ◽  
Galois Covers ◽  
Hyperbolic Riemann Surface

AbstractWe prove a generalized version of Kazhdan’s theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence {\{S_{n}\rightarrow S\}} of finite Galois covers of a hyperbolic Riemann surface S, converging to the universal cover. The theorem states that the sequence of forms on S inherited from the canonical forms on {S_{n}}’s converges uniformly to (a multiple of) the hyperbolic form. We prove a generalized version of this theorem, where the universal cover is replaced with any infinite Galois cover. Along the way, we also prove a Gauss–Bonnet-type theorem in the context of arbitrary infinite Galois covers.

scholarly journals On Level Curves of Harmonic and Analytic Functions on Riemann Surfaces

1969 ◽  
Vol 34 ◽  
pp. 77-87
Author(s):  
Shinji Yamashitad
Keyword(s):  
Riemann Surface ◽  
Subharmonic Function ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Vector Lattice ◽  
Level Curves ◽  
Harmonic Majorant ◽  
Harmonic And Analytic Functions ◽  
Hyperbolic Riemann Surface

In this note we shall denote by R a hyperbolic Riemann surface. Let HP′(R) be the totality of harmonic functions u on R such that every subharmonic function | u | has a harmonic majorant on R. The class HP′(R) forms a vector lattice under the lattice operations:

On a Uniqueness Theorem

1979 ◽  
Vol 31 (5) ◽  
pp. 1072-1076
Author(s):  
Mikio Niimura
Keyword(s):  
Riemann Surface ◽  
Uniqueness Theorem ◽  
Analytic Functions ◽  
Unit Disc ◽  
Riemann Surfaces ◽  
Boundary Behavior ◽  
Uniqueness Theorems ◽  
Ideal Boundary ◽  
Open Riemann Surface ◽  
Hyperbolic Riemann Surfaces

The classical uniqueness theorems of Riesz and Koebe show an important characteristic of boundary behavior of analytic functions in the unit disc. The purpose of this note is to discuss these uniqueness theorems on hyperbolic Riemann surfaces. It will be necessary to give additional hypotheses because Riemann surfaces can be very nasty. So, in this note the Wiener compactification will be used as ideal boundary of Riemann surfaces. The Theorem, Corollaries 1, 2 and 3 are of Riesz type, Riesz-Nevanlinna type, Koebe type and Koebe-Nevanlinna type respectively. Corollaries 4 and 5 are general forms of Corollaries 2 and 3 respectively.Let f be a mapping from an open Riemann surface R into a Riemann surface R′.

scholarly journals The geometry of measured geodesic laminations and measured train tracks

1989 ◽  
Vol 9 (3) ◽  
pp. 587-604 ◽  
Author(s):  
Howard Weiss
Keyword(s):  
Riemann Surface ◽  
Linear Model ◽  
Tangent Space ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Hölder Continuous ◽  
Geodesic Laminations ◽  
Non Linear ◽  
Train Tracks ◽  
Hyperbolic Riemann Surface

AbstractThurston and Kerckhoff have shown that the space of measured geodesic laminations on a hyperbolic Riemann surface serves as a non-linear model of the tangent space to Teichmüller space at the surface. In this paper we show that the natural map between these manifolds has stronger than Hölder continuous regularity.

scholarly journals Almost automorphic solutions of some differential equations in Banach spaces

2000 ◽  
Vol 23 (5) ◽  
pp. 361-365 ◽  
Author(s):  
Gaston Mandata N'Guerekata
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Evolution Equation ◽  
Banach Spaces ◽  
Bounded Solutions ◽  
Almost Automorphic ◽  
Continuous Operators

We discuss the conditions under which bounded solutions of the evolution equationx′(t)=Ax(t)+f(t)in a Banach space are almost automorphic wheneverf(t)is almost automorphic andAgenerates aC0-group of strongly continuous operators. We also give a result for asymptotically almost automorphic solutions for the more general case ofx′(t)=Ax(t)+f(t,x(t)).

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