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

Luděk Zajíček

Displaying similar documents to “A Lipschitz function which is $C^{\infty}$ on a.e. line need not be generically differentiable”

$P^{λ}$ -spaces and $L^{λ}$ -spaces

Nguyen To Nhu (1979)

Colloquium Mathematicae

Similarity:

Lipschitz and uniform embeddings into $ℓ_{\infty}$

N. J. Kalton (2011)

Fundamenta Mathematicae

Similarity:

We show that there is no uniformly continuous selection of the quotient map $Q : ℓ_{\infty} \to ℓ_{\infty} / c ₀$ 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 $B_{X}$ .

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

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 $\mathbb{R}^{n}$ , 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 $M$ .

On continuous composition operators

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 $M_{ψ} : = s u p | | [r, s, t; ψ] | | : r < s < t, r, s, t \in I < \infty$ , 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

Similarity:

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

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 $L i p (α ₁, . . ., α_{N})$ for some $0 < α ₁, . . ., α_{N} < 1$ and its marginal functions satisfy $f (\cdot, x ₂, . . ., x_{N}) \in L i p β ₁, . . ., f (x ₁, . . ., x_{N - 1}, \cdot) \in L i p β_{N}$ for some $0 < β ₁, . . ., β_{N} < 1$ uniformly in the indicated variables $x_{l}$ , 1 ≤ l ≤ N, then $f ̃^{(η ₁, . . ., η_{N})} \in L i p (α ₁, . . ., α_{N})$ for each choice of $(η ₁, . . ., η_{N})$ with $η_{l} = 0$ or 1 for 1 ≤ l ≤ N.

Generalized α-variation and Lebesgue equivalence to differentiable functions

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 $C B V G_{1 / n}$ 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

Similarity:

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.

On the structure of universal differentiability sets

Michael Dymond (2017)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

A subset of $ℝ^{d}$ is called a universal differentiability set if it contains a point of differentiability of every Lipschitz function $f : ℝ^{d} \to ℝ$ . 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.

A Sufficient Condition for the $C^{2}$ -Rectifiability of the Set of Regular Values (in the Sense of Clarke) of a Lipschitz Map

Silvano Delladio (2008)

Bollettino dell'Unione Matematica Italiana

Similarity:

We prove a result about the rectifiability of class $C^{2}$ of the set of regular values (in the sense of Clarke) of a Lipschitz map $\varphi\colon\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}$ with $n<N$

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:

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

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 $ω_{ω}$ 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.

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

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

Archivum Mathematicum

Similarity:

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

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 $C^{p}$ -smooth bump b:X → ℝ such that $| | b^{(p)} {| |}_{\infty}$ 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

Similarity:

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

Similarity:

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

Boundary Value Problems for Higher Order Operators in Lipschitz and $C^{1}$ Domains

Jill Pipher (1992-1993)

Publications mathématiques et informatique de Rennes

Similarity:

Nonlinear mappings preserving at least one eigenvalue

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 $F (x) = u x u^{- 1}$ 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...

Fréchet differentiability via partial Fréchet differentiability

Luděk Zajíček (2023)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let $X_{1}, \dots, X_{n}$ be Banach spaces and $f$ a real function on $X = X_{1} \times \dots \times X_{n}$ . Let $A_{f}$ be the set of all points $x \in X$ at which $f$ is partially Fréchet differentiable but is not Fréchet differentiable. Our results imply that if $X_{1}, \dots, X_{n - 1}$ are Asplund spaces and $f$ is continuous (respectively Lipschitz) on $X$ , then $A_{f}$ is a first category set (respectively a $σ$ -upper porous set). We also prove that if $X$ , $Y$ are separable Banach spaces and $f : X \to Y$ is a Lipschitz mapping, then there exists a $σ$ -upper porous set $A \subset X$ such that $f$ is Fréchet differentiable...

Approximate biflatness and Johnson pseudo-contractibility of some Banach algebras

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

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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.

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

Similarity:

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 $G (x) : = \int_{Ω} g (s, x (s)) d μ (s)$ . 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

Similarity:

By $F_{n} (X)$ , $n \geq 1$ , 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...

Are All Smooth Solutions of Euclidean $S U (2)$ -Yang-Mills Equations Selfdual or Antiselfdual?

R. Flume (1979)

Recherche Coopérative sur Programme n°25

Similarity:

Operator Lipschitz functions on Banach spaces

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 $| | f (B) S - {S f (A) | |}_{(X, Y)} \leq c o n s t | | B S - {S A | |}_{(X, Y)}$ 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 $X = ℓ_{p}$ and $Y = ℓ_{q}$ 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

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. $L^{p}$ , 1 ≤ p ≤ ∞) sense at $x \in ℝ^{m}$ if there are numbers $f_{α} (x)$ , |α| ≤ n, such that $f (x + h) - \sum_{| α | \leq n} f_{α} (x) h^{α} / α!$ 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

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

Interpolation and extension of Lipschitz-Hölder maps on $C_{p}$ spaces

Charles E. C Cleaver (1973)

Colloquium Mathematicae

Similarity:

A new proof of Fréchet differentiability of Lipschitz functions

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 $ℓ_{2}$ (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.

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

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 $f (t x + (1 - t) y) \leq t f (x) + (1 - t) f (y) + m i n [t, (1 - t)] α (| | x - y | |)$ . Then there is a dense $G_{δ}$ -set $A_{G} \subset Ω$ 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

Similarity:

The Cauchy problem for first order system $L (t, x, \partial_{t}, \partial_{x})$ 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

Similarity:

In this note we shall prove that for a continuous function $ϕ : Δ \to ℝ^{n}$ , where $Δ \subset ℝ$ , the paratingent of $ϕ$ at $a \in Δ$ is a non-empty and compact set in $ℝ^{n}$ 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

Similarity:

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 $f \in Λ ⁺_{γ}$ 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) $f / Q \in Λ ⁺_{γ}$ . 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...

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

Displaying similar documents to “A Lipschitz function which is $C^{\infty}$ on a.e. line need not be generically differentiable”