Displaying similar documents to “A Lipschitz function which is C on a.e. line need not be generically differentiable”

Lipschitz and uniform embeddings into

N. J. Kalton (2011)

Fundamenta Mathematicae

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We show that there is no uniformly continuous selection of the quotient map Q : / c 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 .

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 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 .

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.

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 , . . . , α N ) for some 0 < α , . . . , α N < 1 and its marginal functions satisfy f ( · , x , . . . , x N ) L i p β , . . . , f ( x , . . . , x N - 1 , · ) L i p β N for some 0 < β , . . . , β N < 1 uniformly in the indicated variables x l , 1 ≤ l ≤ N, then f ̃ ( η , . . . , η N ) L i p ( α , . . . , α N ) for each choice of ( η , . . . , η N ) with η l = 0 or 1 for 1 ≤ l ≤ N.

Generalized α-variation and Lebesgue equivalence to differentiable functions

Jakub Duda (2009)

Fundamenta Mathematicae

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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 C B V G 1 / n and S B V G 1 / n 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 C B V 1 / n and S B V 1 / n (introduced by Preiss and Laczkovich) for Cⁿ smoothness....

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 and E i are Lipschitz equivalent, and E i and F j are quasi-Lipschitz equivalent when they have the same Hausdorff dimension.

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

Similarity:

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 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 .

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 ( ω ω ) are examples of weakly compactly generated (WCG) Banach spaces which are Lipschitz isomorphic but not isomorphic.

On the range of the derivative of a real-valued function with bounded support

T. Gaspari (2002)

Studia Mathematica

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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 C p -smooth bump b:X → ℝ such that | | b ( p ) | | is finite, then any connected open subset of X* containing 0 is the range of the derivative of a C p -smooth bump. We also study the finite-dimensional case which is quite different....

Canonical Banach function spaces generated by Urysohn universal spaces. Measures as Lipschitz maps

Piotr Niemiec (2009)

Studia Mathematica

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It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure C F L ( r ) of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space r of diameter r, is (isometrically if r = +∞) isomorphic to the space L I P ( r ) of equivalence classes of all real-valued Lipschitz maps on r . The space of all signed (real-valued) Borel measures on r is isometrically embedded in the dual space of C F L ( r ) and it is shown that the image...

On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space

Daniel Azagra, Mar Jiménez-Sevilla (2002)

Bulletin de la Société Mathématique de France

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We study the size of the sets of gradients of bump functions on the Hilbert space 2 , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in 2 can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space 2 can be uniformly approximated by C 1 smooth Lipschitz functions ψ so that the cones generated by the ranges of its derivatives ψ ' ( 2 ) have empty interior. This implies that there are C 1 smooth...

Nonlinear mappings preserving at least one eigenvalue

Constantin Costara, Dušan Repovš (2010)

Studia Mathematica

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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 F ( x ) = u x u - 1 or F ( x ) = u x t u - 1 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...

Generalized gradients for locally Lipschitz integral functionals on non- L p -type spaces of measurable functions

Hôǹg Thái Nguyêñ, Dariusz Pączka (2008)

Banach Center Publications

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Let (Ω,μ) be a measure space, E be an arbitrary separable Banach space, E * ω * 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 G ( x ) : = Ω g ( s , x ( s ) ) d μ ( s ) . Consider the integral functional G defined on some non- L p -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...

Symmetric products of the Euclidean spaces and the spheres

Naotsugu Chinen (2015)

Commentationes Mathematicae Universitatis Carolinae

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By F n ( X ) , n 1 , we denote the n -th symmetric product of a metric space ( X , d ) as the space of the non-empty finite subsets of X with at most n elements endowed with the Hausdorff metric d H . In this paper we shall describe that every isometry from the n -th symmetric product F n ( X ) into itself is induced by some isometry from X into itself, where X is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the n -th symmetric product of the Euclidean space up to bi-Lipschitz equivalence...

Operator Lipschitz functions on Banach spaces

Jan Rozendaal, Fedor Sukochev, Anna Tomskova (2016)

Studia Mathematica

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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 | | f ( B ) S - S f ( A ) | | ( X , Y ) c o n s t | | B S - S A | | ( X , Y ) 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 X = p and Y = q for p < q. We also study the estimate above in the setting of Banach...

Approximate and L p Peano derivatives of nonintegral order

J. Marshall Ash, Hajrudin Fejzić (2005)

Studia Mathematica

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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. L p , 1 ≤ p ≤ ∞) sense at x m if there are numbers f α ( x ) , |α| ≤ n, such that f ( x + h ) - | α | n f α ( x ) h α / α ! is O ( h u ) in the approximate (resp. L p ) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or L p 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...

Filippov Lemma for matrix fourth order differential inclusions

Grzegorz Bartuzel, Andrzej Fryszkowski (2014)

Banach Center Publications

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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 A , B d × d are commutative and the multifunction F : [ 0 , 1 ] × d c l ( d ) is Lipschitz continuous in y with a t-independent constant l < ||A||²||B||². Main theorem. Assume that F : [ 0 , 1 ] × d c l ( d ) i s m e a s u r a b l e i n t a n d i n t e g r a b l y b o u n d e d . L e t y₀ ∈ W4,1 b e a n a r b i t r a r y f u n c t i o n s a t i s f y i n g ( * * ) a n d s u c h t h a t d H ( y ( t ) , F ( t , y ( t ) ) ) p ( t ) a.e. in [0,1], where p₀ ∈ L¹[0,1]. Then there exists a solution y ∈ W4,1 of (*)...

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 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.

An extension of Mazur's theorem on Gateaux differentiability to the class of strongly α (·)-paraconvex functions

S. Rolewicz (2006)

Studia Mathematica

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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 f ( t x + ( 1 - t ) y ) t f ( x ) + ( 1 - t ) f ( y ) + m i n [ t , ( 1 - t ) ] α ( | | x - y | | ) . Then there is a dense G δ -set A G Ω such that f is Gateaux differentiable at every point of A G .

L 2 well-posed Cauchy problems and symmetrizability of first order systems

Guy Métivier (2014)

Journal de l’École polytechnique — Mathématiques

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The Cauchy problem for first order system L ( t , x , t , x ) is known to be well-posed in L 2 when it admits a microlocal symmetrizer S ( t , x , ξ ) which is smooth in ξ and Lipschitz continuous in ( t , x ) . This paper contains three main results. First we show that a Lipschitz smoothness globally in ( t , x , ξ ) 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...

On compactness and connectedness of the paratingent

Wojciech Zygmunt (2016)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this note we shall prove that for a continuous function ϕ : Δ n , where Δ ,  the paratingent of ϕ at a Δ is a non-empty and compact set in n if and only if ϕ satisfies Lipschitz condition in a neighbourhood of a . Moreover, in this case the paratingent is a connected set.

Ideals in big Lipschitz algebras of analytic functions

Thomas Vils Pedersen (2004)

Studia Mathematica

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For 0 < γ ≤ 1, let Λ γ 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 J γ ( E , Q ) be the closed ideal in Λ γ consisting of those functions f Λ γ for which (i) f = 0 on E, (ii) | f ( z ) - f ( w ) | = o ( | z - w | γ ) as d(z,E),d(w,E) → 0, (iii) f / Q Λ γ . Also, for a closed ideal I in Λ γ , let E I = z ∈ : f(z) = 0 for every f ∈ I and let Q I be the greatest common divisor of the inner parts of non-zero functions...

Z k -actions with a special fixed point set

Pedro L. Q. Pergher, Rogério de Oliveira (2005)

Fundamenta Mathematicae

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Let Fⁿ be a connected, smooth and closed n-dimensional manifold satisfying the following property: if N m is any smooth and closed m-dimensional manifold with m > n and T : N m N m is a smooth involution whose fixed point set is Fⁿ, then m = 2n. We describe the equivariant cobordism classification of smooth actions ( M m ; Φ ) of the group G = Z k on closed smooth m-dimensional manifolds M m for which the fixed point set of the action is a submanifold Fⁿ with the above property. This generalizes a result of F....

Smoothness of Green's functions and Markov-type inequalities

Leokadia Białas-Cież (2011)

Banach Center Publications

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Let E be a compact set in the complex plane, g E be the Green function of the unbounded component of E with pole at infinity and M ( E ) = s u p ( | | P ' | | E ) / ( | | P | | E ) where the supremum is taken over all polynomials P | E 0 of degree at most n, and | | f | | E = s u p | f ( z ) | : z E . The paper deals with recent results concerning a connection between the smoothness of g E (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence M ( E ) n = 1 , 2 , . . . . Some additional conditions are given for special classes of sets.

Remarks on the Bourgain-Brezis-Mironescu Approach to Sobolev Spaces

B. Bojarski (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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For a function f L l o c p ( ) the notion of p-mean variation of order 1, p ( f , ) 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 W 1 , p ( ) in terms of p ( f , ) is directly related to the characterisation of W 1 , p ( ) by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.

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 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. ...