-spaces and -spaces
Nguyen To Nhu (1979)
Colloquium Mathematicae
Similarity:
Nguyen To Nhu (1979)
Colloquium Mathematicae
Similarity:
N. J. Kalton (2011)
Fundamenta Mathematicae
Similarity:
We show that there is no uniformly continuous selection of the quotient map 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 onto .
Alberto Bressan, Agostino Cortesi (1986)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
Similarity:
Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz , definita su un sottoinsieme di uno spazio di Hilbert a valori compatti e convessi in , può essere estesa su tutto ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz .
Wilhelmina Smajdor (2010)
Annales Polonici Mathematici
Similarity:
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 , 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...
Vanda Fülöp (2006)
Colloquium Mathematicae
Similarity:
Denote by the sum of a double sine series with nonnegative coefficients. We present necessary and sufficient coefficient conditions in order that 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.
Ferenc Móricz (2009)
Colloquium Mathematicae
Similarity:
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 for some and its marginal functions satisfy for some uniformly in the indicated variables , 1 ≤ l ≤ N, then for each choice of with or 1 for 1 ≤ l ≤ N.
Jakub Duda (2009)
Fundamenta Mathematicae
Similarity:
We find conditions on a real function f:[a,b] → ℝ equivalent to being Lebesgue equivalent to an n-times differentiable function (n ≥ 2); a simple solution in the case n = 2 appeared in an earlier paper. For that purpose, we introduce the notions of and functions, which play analogous rôles for the nth order differentiability to the classical notion of a VBG⁎ function for the first order differentiability, and the classes and (introduced by Preiss and Laczkovich) for Cⁿ smoothness....
Ying Xiong, Lifeng Xi (2009)
Studia Mathematica
Similarity:
This paper studies the geometric structure of graph-directed sets from the point of view of Lipschitz equivalence. It is proved that if and are dust-like graph-directed sets satisfying the transitivity condition, then and are Lipschitz equivalent, and and are quasi-Lipschitz equivalent when they have the same Hausdorff dimension.
Silvano Delladio (2008)
Bollettino dell'Unione Matematica Italiana
Similarity:
We prove a result about the rectifiability of class of the set of regular values (in the sense of Clarke) of a Lipschitz map with
Alberto Bressan, Agostino Cortesi (1986)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
Similarity:
Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz , definita su un sottoinsieme di uno spazio di Hilbert a valori compatti e convessi in , può essere estesa su tutto ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz .
Witold Marciszewski (2003)
Studia Mathematica
Similarity:
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 and with the third derived set empty such that the space C(K) is not isomorphic to any c₀(Γ). For this compactum K, the spaces C(K) and are examples of weakly compactly generated (WCG) Banach spaces which are Lipschitz isomorphic but not isomorphic.
T. Gaspari (2002)
Studia Mathematica
Similarity:
We study the set f’(X) = f’(x): x ∈ X when f:X → ℝ is a differentiable bump. We first prove that for any C²-smooth bump f: ℝ² → ℝ the range of the derivative of f must be the closure of its interior. Next we show that if X is an infinite-dimensional separable Banach space with a -smooth bump b:X → ℝ such that is finite, then any connected open subset of X* containing 0 is the range of the derivative of a -smooth bump. We also study the finite-dimensional case which is quite different....
Piotr Niemiec (2009)
Studia Mathematica
Similarity:
It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space of diameter r, is (isometrically if r = +∞) isomorphic to the space of equivalence classes of all real-valued Lipschitz maps on . The space of all signed (real-valued) Borel measures on is isometrically embedded in the dual space of and it is shown that the image...
Daniel Azagra, Mar Jiménez-Sevilla (2002)
Bulletin de la Société Mathématique de France
Similarity:
We study the size of the sets of gradients of bump functions on the Hilbert space , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space can be uniformly approximated by smooth Lipschitz functions so that the cones generated by the ranges of its derivatives have empty interior. This implies that there are smooth...
Jill Pipher (1992-1993)
Publications mathématiques et informatique de Rennes
Similarity:
Constantin Costara, Dušan Repovš (2010)
Studia Mathematica
Similarity:
We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either or for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying...
Hôǹg Thái Nguyêñ, Dariusz Pączka (2008)
Banach Center Publications
Similarity:
Let (Ω,μ) be a measure space, E be an arbitrary separable Banach space, be the dual equipped with the weak* topology, and g:Ω × E → ℝ be a Carathéodory function which is Lipschitz continuous on each ball of E for almost all s ∈ Ω. Put . Consider the integral functional G defined on some non--type Banach space X of measurable functions x: Ω → E. We present several general theorems on sufficient conditions under which any element γ ∈ X* of Clarke’s generalized gradient (multivalued...
Naotsugu Chinen (2015)
Commentationes Mathematicae Universitatis Carolinae
Similarity:
By , , we denote the -th symmetric product of a metric space as the space of the non-empty finite subsets of with at most elements endowed with the Hausdorff metric . In this paper we shall describe that every isometry from the -th symmetric product into itself is induced by some isometry from into itself, where is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the -th symmetric product of the Euclidean space up to bi-Lipschitz equivalence...
R. Flume (1979)
Recherche Coopérative sur Programme n°25
Similarity:
Jan Rozendaal, Fedor Sukochev, Anna Tomskova (2016)
Studia Mathematica
Similarity:
Let X, Y be Banach spaces and let (X,Y) be the space of bounded linear operators from X to Y. We develop the theory of double operator integrals on (X,Y) and apply this theory to obtain commutator estimates of the form for a large class of functions f, where A ∈ (X), B ∈ (Y) are scalar type operators and S ∈ (X,Y). In particular, we establish this estimate for f(t): = |t| and for diagonalizable operators on and for p < q. We also study the estimate above in the setting of Banach...
J. Marshall Ash, Hajrudin Fejzić (2005)
Studia Mathematica
Similarity:
Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. , 1 ≤ p ≤ ∞) sense at if there are numbers , |α| ≤ n, such that is in the approximate (resp. ) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and...
Grzegorz Bartuzel, Andrzej Fryszkowski (2014)
Banach Center Publications
Similarity:
In the paper we give an analogue of the Filippov Lemma for the fourth order differential inclusions y = y”” - (A² + B²)y” + A²B²y ∈ F(t,y), (*) with the initial conditions y(0) = y’(0) = y”(0) = y”’(0) = 0, (**) where the matrices are commutative and the multifunction is Lipschitz continuous in y with a t-independent constant l < ||A||²||B||². Main theorem. Assume that y₀ ∈ W4,1 a.e. in [0,1], where p₀ ∈ L¹[0,1]. Then there exists a solution y ∈ W4,1 of (*)...
Charles E. C Cleaver (1973)
Colloquium Mathematicae
Similarity:
Joram Lindenstrauss, David Preiss (2000)
Journal of the European Mathematical Society
Similarity:
We give a relatively simple (self-contained) proof that every real-valued Lipschitz function on (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 closure of the set of its points of Gâteaux differentiability is norm separable.
S. Rolewicz (2006)
Studia Mathematica
Similarity:
Let (X,||·||) be a separable real Banach space. Let f be a real-valued strongly α(·)-paraconvex function defined on an open convex subset Ω ⊂ X, i.e. such that . Then there is a dense -set such that f is Gateaux differentiable at every point of .
Guy Métivier (2014)
Journal de l’École polytechnique — Mathématiques
Similarity:
The Cauchy problem for first order system is known to be well-posed in when it admits a microlocal symmetrizer which is smooth in and Lipschitz continuous in . This paper contains three main results. First we show that a Lipschitz smoothness globally in is sufficient. Second, we show that the existence of symmetrizers with a given smoothness is equivalent to the existence of having the same smoothness. This notion was first introduced in []. This is the key point to prove...
Wojciech Zygmunt (2016)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
Similarity:
In this note we shall prove that for a continuous function , where , the paratingent of at is a non-empty and compact set in if and only if satisfies Lipschitz condition in a neighbourhood of . Moreover, in this case the paratingent is a connected set.
Thomas Vils Pedersen (2004)
Studia Mathematica
Similarity:
For 0 < γ ≤ 1, let be the big Lipschitz algebra of functions analytic on the open unit disc which satisfy a Lipschitz condition of order γ on ̅. For a closed set E on the unit circle and an inner function Q, let be the closed ideal in consisting of those functions for which (i) f = 0 on E, (ii) as d(z,E),d(w,E) → 0, (iii) . Also, for a closed ideal I in , let = z ∈ : f(z) = 0 for every f ∈ I and let be the greatest common divisor of the inner parts of non-zero functions...
Pedro L. Q. Pergher, Rogério de Oliveira (2005)
Fundamenta Mathematicae
Similarity:
Let Fⁿ be a connected, smooth and closed n-dimensional manifold satisfying the following property: if is any smooth and closed m-dimensional manifold with m > n and is a smooth involution whose fixed point set is Fⁿ, then m = 2n. We describe the equivariant cobordism classification of smooth actions of the group on closed smooth m-dimensional manifolds for which the fixed point set of the action is a submanifold Fⁿ with the above property. This generalizes a result of F....
Leokadia Białas-Cież (2011)
Banach Center Publications
Similarity:
Let E be a compact set in the complex plane, be the Green function of the unbounded component of with pole at infinity and where the supremum is taken over all polynomials of degree at most n, and . The paper deals with recent results concerning a connection between the smoothness of (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence . Some additional conditions are given for special classes of sets.
B. Bojarski (2011)
Bulletin of the Polish Academy of Sciences. Mathematics
Similarity:
For a function the notion of p-mean variation of order 1, is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space in terms of is directly related to the characterisation of by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.
Matías Raja (2014)
Studia Mathematica
Similarity:
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 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. ...
A. Pełczyński, H. Rosenthal (1975)
Studia Mathematica
Similarity:
D. W. Hajek, I. Irizarry (1981)
Matematički Vesnik
Similarity: