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1.
The main purpose of this article is to generalize a recent result about isometries of Lipschitz spaces. Botelho, Fleming and Jamison [2] described surjective linear isometries between vector-valued Lipschitz spaces under certain conditions. In this article, we extend the main result of [2] by removing the quasi-sub-reflexivity condition from Banach spaces. 相似文献
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Fernanda Botelho James E. Jamison 《Journal of Mathematical Analysis and Applications》2011,381(2):821-832
For a Banach space E and a compact metric space (X,d), a function F:X→E is a Lipschitz function if there exists k>0 such that
5.
E. Durand-Cartagena 《Journal of Mathematical Analysis and Applications》2010,363(2):525-548
For a metric space X, we study the space D∞(X) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D∞(X) is compared with the space LIP∞(X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D∞(X) with the Newtonian-Sobolev space N1,∞(X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D∞(X)=N1,∞(X). 相似文献
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Let (X, d) be a compact metric and 0 < α < 1. The space Lip α (X) of Hölder functions of order α is the Banach space of all functions ? from X into \(\mathbb{K}\) such that ∥?∥ = max{∥?∥∞, L(?)} < ∞, where is the Hölder seminorm of ?. The closed subspace of functions ? such that is denoted by lip α (X). We determine the form of all bijective linear maps from lip α (X) onto lip α (Y) that preserve the Hölder seminorm.
相似文献
$L(f) = sup\{ \left| {f(x) - f(y)} \right|/d^\alpha (x,y):x,y \in X, x \ne y\} $
$\mathop {\lim }\limits_{d(x,y) \to 0} \left| {f(x) - f(y)} \right|/d^\alpha (x,y) = 0$
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A. Jiménez-Vargas 《Journal of Mathematical Analysis and Applications》2008,337(2):984-993
Given a real number α∈(0,1) and a metric space (X,d), let Lipα(X) be the algebra of all scalar-valued bounded functions f on X such that
10.
In this paper we study multilinear isometries defined on certain subspaces of vector-valued continuous functions. We provide conditions under which such maps can be properly represented. Our results contain all known results concerning linear and bilinear isometries defined between spaces of continuous functions. The key result is a vector-valued version of the additive Bishop’s Lemma, which we think has interest in itself. 相似文献
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Let X be a compact metric space and let Lip(X) be the Banach algebra of all scalar- valued Lipschitz functions on X, endowed with a natural norm. For each f ∈ Lip(X), σπ(f) denotes the peripheral spectrum of f. We state that any map Φ from Lip(X) onto Lip(Y) which preserves multiplicatively the peripheral spectrum:
σπ(Φ(f)Φ(g)) = σπ(fg), A↓f, g ∈ Lip(X)
is a weighted composition operator of the form Φ(f) = τ· (f °φ) for all f ∈ Lip(X), where τ : Y → {-1, 1} is a Lipschitz function and φ : Y→ X is a Lipschitz homeomorphism. As a consequence of this result, any multiplicatively spectrum-preserving surjective map between Lip(X)-algebras is of the form above. 相似文献
σπ(Φ(f)Φ(g)) = σπ(fg), A↓f, g ∈ Lip(X)
is a weighted composition operator of the form Φ(f) = τ· (f °φ) for all f ∈ Lip(X), where τ : Y → {-1, 1} is a Lipschitz function and φ : Y→ X is a Lipschitz homeomorphism. As a consequence of this result, any multiplicatively spectrum-preserving surjective map between Lip(X)-algebras is of the form above. 相似文献
13.
If f(z) =Σ∞ n=0 anzn and g(z) =Σ∞n=0bnzn for functions f, g are analytic in the unit disc, the Hadamard products of f and g is defined by f * g = ∞ n=0 a n b n z n . In this paper, the Lipschitz spaces Λ(s, α) and QK type spaces are studied in terms of the Hadamard products. 相似文献
14.
Fernanda Botelho James Jamison Antonio Jiménez-Vargas 《Journal of Mathematical Analysis and Applications》2012,386(2):910-920
We characterize projections on spaces of Lipschitz functions expressed as the average of two and three linear surjective isometries. Generalized bi-circular projections are the only projections on these spaces given as the convex combination of two surjective isometries. 相似文献
15.
《Mathematische Nachrichten》2018,291(11-12):1899-1907
In this article, we describe isometries over the Lipschitz spaces under certain conditions. Indeed, we provide a unified proof for the main results of 3 and 5 in a more general setting. Finally, we extend our results for some other functions spaces like the space of vector‐valued little Lipschitz maps and pointwise Lipschitz maps. 相似文献
16.
Zhijian Wu 《Frontiers of Mathematics in China》2007,2(2):287-303
We characterize the symbol functions so that the associated commutators with symbol functions and the Hilbert transform are
bounded on Lipschitz space Λ
α
p
, where 1 < p < ∞ and 0 < α < 1/p. Properties of such symbols are also discussed.
相似文献
17.
We study the numerical radius of Lipschitz operators on Banach spaces via the Lipschitz numerical index, which is an analogue of the numerical index in Banach space theory. We give a characterization of the numerical radius and obtain a necessary and sufficient condition for Banach spaces to have Lipschitz numerical index 1. As an application, we show that real lush spaces and C -rich subspaces have Lipschitz numerical index 1. Moreover, using the Gâteaux differentiability of Lipschitz operators, we characterize the Lipschitz numerical index of separable Banach spaces with the RNP. Finally, we prove that the Lipschitz numerical index has the stability properties for the c0-, l1-, and l∞-sums of spaces and vector-valued function spaces. From this, we show that the C(K) spaces, L1(μ)-spaces and L∞(ν)-spaces have Lipschitz numerical index 1. 相似文献
18.
Yashan Xu 《Journal of Mathematical Analysis and Applications》2007,327(1):665-678
A uniformly k-Lipschitz feedback optimal control problem is considered in a linear quadratic framework. The value function is derived by a comparison-based reasoning, via which a necessary condition to the existence of optimal solutions is obtained: optimal feedback controls must be linear. 相似文献
19.
Summary A real valued function <InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource
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defined on a real interval $I$ is called \emph{$d$-Lipschitz} if it satisfies $|\ell(x)- \ell(y)| \le d(x,y)$ for $x,y\in
I$. In this paper, we investigate when a function $p\: I \to \bR$ can be decomposed in the form $p=q+ \ell$, where $q$ is
increasing and $\ell$ is $d$-Lipschitz. In the general case when $d\: I^{2} \to \bR$ is an arbitrary semimetric, a function
$p\: I \to \bR$ can be written in the form $p=q+ \ell$ if and only if \vspace{-4pt} <InlineEquation ID=IE"1"><EquationSource
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\sum_{i=1}^{n}{\big(p(s_{i})-p(t_{i})-d(t_{i},s_{i}) \big)^{+}} \le \sum_{j=1}^{m}{\big(p(v_{j})-p(u_{j})+d(u_{j},v_{j}) \big)}
\vspace{-4pt} $$ is fulfilled for all real numbers $t_{1}<s_{1}, \dots, t_{n}<s_{n}$ and $u_{1}<v_{1}, \dots, u_{m}<v_{m}$
in $I$ satisfying the condition \vspace{-4pt} $$ \sum_{i=1}^{n} 1_{\left]t_i,s_i\right]}= \sum_{j=1}^{m} 1_{\left]u_j,v_j\right]},
\vspace{-4pt} $$ where $1_{\left]a,b\right]}$ denotes the characteristic function of the interval $\left]a,b\right]$. In the
particular case when $d\: I^{2} \to R$ is a so-called concave semimetric, a function $p\: I \to \bR$ is of the form $p=q+
\ell$ if and only if \vspace{-4pt} $$ 0 \le \sum_{k=1}^{n}{d(x_{2k-1},x_{2k})} + d(x_0,x_{2n+1}) + \sum_{k=0}^{n}{\big(p(x_{2k+1})-p(x_{2k})\big)}
\vspace{-4pt} $$ holds for all $x_0\le x_1\ki \cdots\ki x_{2n}\le x_{2n+1}$ in $I$. 相似文献
20.
Miroslav Pavlović 《Mathematische Zeitschrift》2008,258(1):81-86
We prove that maps into
if and only if belongs to . In the case β < 1, we give another two equivalent conditions.
Supported by MNZŽS Serbia, Project No. ON144010. 相似文献