scholarly journals Smooth Approximation of Lipschitz Functions on Finsler Manifolds

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
M. I. Garrido ◽  
J. A. Jaramillo ◽  
Y. C. Rangel
Keyword(s):  
Lipschitz Function ◽  
Composition Operators ◽  
Lipschitz Functions ◽  
Finsler Manifold ◽  
Positive Continuous Function ◽  
Smooth Approximation ◽  
Finsler Manifolds ◽  
Lipschitz Constants ◽  
Bounded Derivative

We study the smooth approximation of Lipschitz functions on Finsler manifolds, keeping control on the corresponding Lipschitz constants. We prove that, given a Lipschitz functionf:M→ℝdefined on a connected, second countable Finsler manifoldM, for each positive continuous functionε:M→(0,∞)and eachr>0, there exists aC1-smooth Lipschitz functiong:M→ℝsuch that|f(x)-g(x)|≤ε(x), for everyx∈M, andLip(g)≤Lip(f)+r. As a consequence, we derive a completeness criterium in the class of what we call quasi-reversible Finsler manifolds. Finally, considering the normed algebraCb1(M)of allC1functions with bounded derivative on a complete quasi-reversible Finsler manifoldM, we obtain a characterization of algebra isomorphismsT:Cb1(N)→Cb1(M)as composition operators. From this we obtain a variant of Myers-Nakai Theorem in the context of complete reversible Finsler manifolds.

Composition operators on μ-Bloch spaces

2009 ◽  
Vol 61 (1) ◽  
pp. 50-75 ◽  
Author(s):  
Huaihui Chen ◽  
Paul Gauthier
Keyword(s):  
Continuous Function ◽  
Unit Ball ◽  
Composition Operator ◽  
Composition Operators ◽  
Sufficient Conditions ◽  
Positive Continuous Function ◽  
Bloch Spaces ◽  
Bloch Functions ◽  
Necessary And Sufficient

Abstract. Given a positive continuous function μ on the interval 0 < t ≤ 1, we consider the space of so-called μ-Bloch functions on the unit ball. If μ(t ) = t, these are the classical Bloch functions. For μ, we define a metric Fμz (u) in terms of which we give a characterization of μ-Bloch functions. Then, necessary and sufficient conditions are obtained in order that a composition operator be a bounded or compact operator between these generalized Bloch spaces. Our results extend those of Zhang and Xiao.

scholarly journals A Geometric characterization of Finsler manifolds of constant curvatureK=1

2000 ◽  
Vol 23 (6) ◽  
pp. 399-407 ◽  
Author(s):  
A. Bejancu ◽  
H. R. Farran
Keyword(s):  
Vector Field ◽  
Constant Curvature ◽  
Killing Vector ◽  
Finsler Manifold ◽  
Geometric Characterization ◽  
Killing Vector Field ◽  
Finsler Manifolds

We prove that a Finsler manifold&#x1D53D;mis of constant curvatureK=1if and only if the unit horizontal Liouville vector field is a Killing vector field on the indicatrix bundleIMof&#x1D53D;m.

scholarly journals Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

2021 ◽  
Author(s):  
Bin Liu ◽  
Jouni Rättyä ◽  
Fanglei Wu
Keyword(s):  
Bergman Space ◽  
Open Problem ◽  
Lebesgue Space ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Carleson Measures ◽  
Doubling Weights ◽  
The Way

AbstractBounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.

scholarly journals Invertible Weighted Composition Operators on the Fock Space ofCN

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Liankuo Zhao
Keyword(s):  
Fock Space ◽  
Composition Operators ◽  
Complete Characterization ◽  
Weighted Composition Operators ◽  
Weighted Composition

We give a complete characterization of bounded invertible weighted composition operators on the Fock space ofCN.

scholarly journals Geodesic Random Walks, Diffusion Processes and Brownian Motion on Finsler Manifolds

2021 ◽  
Author(s):  
Tianyu Ma ◽  
Vladimir S. Matveev ◽  
Ilya Pavlyukevich
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Random Walks ◽  
Diffusion Processes ◽  
Riemannian Metric ◽  
Finsler Manifold ◽  
Finsler Manifolds ◽  
Bounded Geometry ◽  
And Brownian Motion

AbstractWe show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.

scholarly journals Isometric and Closed-Range Composition Operators between Bloch-Type Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-15
Author(s):  
Nina Zorboska
Keyword(s):  
Holomorphic Function ◽  
Composition Operators ◽  
Function Spaces ◽  
Complete Characterization ◽  
Closed Range ◽  
Different Types ◽  
Type Spaces ◽  
Range Determination

We present an overview of the known results describing the isometric and closed-range composition operators on different types of holomorphic function spaces. We add new results and give a complete characterization of the isometric univalently induced composition operators acting between Bloch-type spaces. We also add few results on the closed-range determination of composition operators on Bloch-type spaces and present the problems that are still open.

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