Displaying similar documents to “Biseparating maps on generalized Lipschitz spaces”

A new proof of Fréchet differentiability of Lipschitz functions

Joram Lindenstrauss, David Preiss (2000)

Journal of the European Mathematical Society

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We give a relatively simple (self-contained) proof that every real-valued Lipschitz function on ℓ 2 (or more generally on an Asplund space) has points of Fréchet differentiability. Somewhat more generally, we show that a real-valued Lipschitz function on a separable Banach space has points of Fréchet differentiability provided that the w * closure of the set of its points of Gâteaux differentiability is norm separable.

Some algebraic and homological properties of Lipschitz algebras and their second duals

F. Abtahi, E. Byabani, A. Rejali (2019)

Archivum Mathematicum

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Let ( X , d ) be a metric space and α > 0 . We study homological properties and different types of amenability of Lipschitz algebras Lip α X and their second duals. Precisely, we first provide some basic properties of Lipschitz algebras, which are important for metric geometry to know how metric properties are reflected in simple properties of Lipschitz functions. Then we show that all of these properties are equivalent to either uniform discreteness or finiteness of X . Finally, some results concerning...

On continuous composition operators

Wilhelmina Smajdor (2010)

Annales Polonici Mathematici

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Let I ⊂ ℝ be an interval, Y be a normed linear space and Z be a Banach space. We investigate the Banach space Lip₂(I,Z) of all functions ψ: I → Z such that M ψ : = s u p | | [ r , s , t ; ψ ] | | : r < s < t , r , s , t ∈ I < ∞ , where [r,s,t;ψ]:= ((s-r)ψ(t)+(t-s)ψ(r)-(t-r)ψ(s))/((t-r)(t-s)(s-r)). We show that ψ ∈ Lip₂(I,Z) if and only if ψ is differentiable and its derivative ψ’ is Lipschitzian. Suppose the composition operator N generated by h: I × Y → Z, (Nφ)(t):= h(t,φ(t)), maps the set (I,Y) of all affine functions φ: I → Y into Lip₂(I,Z). We prove...

Double sine series with nonnegative coefficients and Lipschitz classes

Vanda Fülöp (2006)

Colloquium Mathematicae

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Denote by f s s ( x , y ) the sum of a double sine series with nonnegative coefficients. We present necessary and sufficient coefficient conditions in order that f s s belongs to the two-dimensional multiplicative Lipschitz class Lip(α,β) for some 0 < α ≤ 1 and 0 < β ≤ 1. Our theorems are extensions of the corresponding theorems by Boas for single sine series.

Best constants for Lipschitz embeddings of metric spaces into c₀

N. J. Kalton, G. Lancien (2008)

Fundamenta Mathematicae

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We answer a question of Aharoni by showing that every separable metric space can be Lipschitz 2-embedded into c₀ and this result is sharp; this improves earlier estimates of Aharoni, Assouad and Pelant. We use our methods to examine the best constant for Lipschitz embeddings of the classical ℓ p -spaces into c₀ and give other applications. We prove that if a Banach space embeds almost isometrically into c₀, then it embeds linearly almost isometrically into c₀. We also study Lipschitz embeddings...

A Lipschitz function which is C ∞ on a.e. line need not be generically differentiable

Luděk Zajíček (2013)

Colloquium Mathematicae

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We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is C ∞ smooth on a.e. line parallel to v and f is Gâteaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dimX > 1) is an arbitrary Banach space and “a.e.” has any usual “measure sense”. This example gives an answer to a natural question concerning the author’s recent study of linearly essentially smooth functions (which generalize essentially...

Lipschitz and uniform embeddings into ℓ ∞

N. J. Kalton (2011)

Fundamenta Mathematicae

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We show that there is no uniformly continuous selection of the quotient map Q : ℓ ∞ → ℓ ∞ / c ₀ relative to the unit ball. We use this to construct an answer to a problem of Benyamini and Lindenstrauss; there is a Banach space X such that there is a no Lipschitz retraction of X** onto X; in fact there is no uniformly continuous retraction from B X * * onto B X .

On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function

Luděk Zajíček (1997)

Commentationes Mathematicae Universitatis Carolinae

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We improve a theorem of P.G. Georgiev and N.P. Zlateva on Gâteaux differentiability of Lipschitz functions in a Banach space which admits a Lipschitz uniformly Gâteaux differentiable bump function. In particular, our result implies the following theorem: If d is a distance function determined by a closed subset A of a Banach space X with a uniformly Gâteaux differentiable norm, then the set of points of X ∖ A at which d is not Gâteaux differentiable is not only a first category set, but...

On the structure of universal differentiability sets

Michael Dymond (2017)

Commentationes Mathematicae Universitatis Carolinae

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A subset of ℝ d is called a universal differentiability set if it contains a point of differentiability of every Lipschitz function f : ℝ d → ℝ . We show that any universal differentiability set contains a ‘kernel’ in which the points of differentiability of each Lipschitz function are dense. We further prove that no universal differentiability set may be decomposed as a countable union of relatively closed, non-universal differentiability sets.

Lipschitz extensions of convex-valued maps

Alberto Bressan, Agostino Cortesi (1986)

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

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Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz M , definita su un sottoinsieme di uno spazio di Hilbert H a valori compatti e convessi in ℝ n , può essere estesa su tutto H ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz M .

Lipschitz equivalence of graph-directed fractals

Ying Xiong, Lifeng Xi (2009)

Studia Mathematica

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This paper studies the geometric structure of graph-directed sets from the point of view of Lipschitz equivalence. It is proved that if E i i and F j j are dust-like graph-directed sets satisfying the transitivity condition, then E i ₁ and E i ₂ are Lipschitz equivalent, and E i and F j are quasi-Lipschitz equivalent when they have the same Hausdorff dimension.

Approximate biflatness and Johnson pseudo-contractibility of some Banach algebras

Amir Sahami, Mohammad R. Omidi, Eghbal Ghaderi, Hamzeh Zangeneh (2020)

Commentationes Mathematicae Universitatis Carolinae

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We study the structure of Lipschitz algebras under the notions of approximate biflatness and Johnson pseudo-contractibility. We show that for a compact metric space X , the Lipschitz algebras Lip α ( X ) and lip α ( X ) are approximately biflat if and only if X is finite, provided that 0 < α < 1 . We give a necessary and sufficient condition that a vector-valued Lipschitz algebras is Johnson pseudo-contractible. We also show that some triangular Banach algebras are not approximately biflat.

Two applications of smoothness in C(K) spaces

Matías Raja (2014)

Studia Mathematica

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A simple observation about embeddings of smooth Banach spaces into C(K) spaces allows us to construct a parametrization of the separable Banach spaces using closed subsets of the interval [0,1]. The same idea is applied to the study of the isometric embedding of ℓ p spaces into certain C(K) spaces with the additional condition that the functions of the image must be Lipschitz with respect to a fixed finer metric on K. The feasibility of that kind of embeddings is related to Szlenk indices. ...

Multiple conjugate functions and multiplicative Lipschitz classes

Ferenc Móricz (2009)

Colloquium Mathematicae

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We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function f belongs to the multiplicative Lipschitz class L i p ( α ₁ , . . . , α N ) for some 0 < α ₁ , . . . , α N < 1 and its marginal functions satisfy f ( · , x ₂ , . . . , x N ) ∈ L i p β ₁ , . . . , f ( x ₁ , . . . , x N - 1 , · ) ∈ L i p β N for some 0 < β ₁ , . . . , β N < 1 uniformly in the indicated variables x l , 1 ≤ l ≤ N, then f ̃ ( η ₁ , . . . , η N ) ∈ L i p ( α ₁ , . . . , α N ) for each choice of ( η ₁ , . . . , η N ) with η l = 0 or 1 for 1 ≤ l ≤ N.

On Banach spaces C(K) isomorphic to c₀(Γ)

Witold Marciszewski (2003)

Studia Mathematica

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We give a characterization of compact spaces K such that the Banach space C(K) is isomorphic to the space c₀(Γ) for some set Γ. As an application we show that there exists an Eberlein compact space K of weight ω ω and with the third derived set K ( 3 ) empty such that the space C(K) is not isomorphic to any c₀(Γ). For this compactum K, the spaces C(K) and c ₀ ( ω ω ) are examples of weakly compactly generated (WCG) Banach spaces which are Lipschitz isomorphic but not isomorphic.