scholarly journals Flattening functions on flowers

2007 ◽  
Vol 27 (6) ◽  
pp. 1865-1886 ◽  
Author(s):  
EDMUND HARRISS ◽  
OLIVER JENKINSON
Keyword(s):  
Lipschitz Function ◽  
Linear Constraints ◽  
Lipschitz Functions ◽  
Jump Discontinuity ◽  
Connected Components ◽  
Expanding Map ◽  
Maximizing Measure

AbstractLet T be an orientation-preserving Lipschitz expanding map of the circle ${\mathbb T}$. A pre-image selector is a map $\tau : {\mathbb T} \to {\mathbb T}$ with finitely many discontinuities, each of which is a jump discontinuity, and such that τ(x)∈T−1(x) for all $x\in {\mathbb T}$. The closure of the image of a pre-image selector is called a flower and a flower with p connected components is called a p-flower. We say that a real-valued Lipschitz function can be Lipschitz flattened on a flower whenever it is Lipschitz cohomologous to a constant on that flower. The space of Lipschitz functions which can be flattened on a given p-flower is shown to be of codimension p in the space of all Lipschitz functions, and the linear constraints determining this subspace are derived explicitly. If a Lipschitz function f has a maximizing measure S which is Sturmian (i.e. is carried by a 1-flower), it is shown that f can be Lipschitz flattened on some 1-flower carrying S.

scholarly journals Lipschitz functions with unexpectedly large sets of nondifferentiability points

2005 ◽  
Vol 2005 (4) ◽  
pp. 361-373 ◽  
Author(s):  
Marianna Csörnyei ◽  
David Preiss ◽  
Jaroslav Tišer
Keyword(s):  
Lipschitz Function ◽  
Measure Zero ◽  
Common Point ◽  
Lipschitz Functions ◽  
Large Sets ◽  
Set Of Points ◽  
Dense Set

It is known that everyGδsubsetEof the plane containing a dense set of lines, even if it has measure zero, has the property that every real-valued Lipschitz function onℝ2has a point of differentiability inE. Here we show that the set of points of differentiability of Lipschitz functions inside such sets may be surprisingly tiny: we construct aGδsetE⊂ℝ2containing a dense set of lines for which there is a pair of real-valued Lipschitz functions onℝ2having no common point of differentiability inE, and there is a real-valued Lipschitz function onℝ2whose set of points of differentiability inEis uniformly purely unrectifiable.

scholarly journals Approximation of functions by means of Lipschitz functions

1963 ◽  
Vol 3 (2) ◽  
pp. 134-150 ◽  
Author(s):  
J. H. Michael
Keyword(s):  
Lipschitz Function ◽  
Unit Cube ◽  
Lipschitz Functions ◽  
Approximation Of Functions ◽  
Image Position ◽  
Elementary Area

Let Q denote the closed unit cube in Rn. The elementary area A(f) of a Lipschitz function f on Q is given by the formula.

scholarly journals An Integralgeometric Approach to Dorronsoro Estimates

2019 ◽  
Author(s):  
Tuomas Orponen
Keyword(s):  
Lipschitz Function ◽  
Parabolic Problem ◽  
Elementary Proof ◽  
Lipschitz Functions ◽  
Geometric Proof ◽  
Proof Technique ◽  
Almost Everywhere ◽  
Small Scales ◽  
Affine Functions ◽  
Similar Proof

Abstract A theorem of Dorronsoro from 1985 quantifies the fact that a Lipschitz function $f \colon \mathbb{R}^{n} \to \mathbb{R}$ can be approximated by affine functions almost everywhere, and at sufficiently small scales. This paper contains a new, purely geometric, proof of Dorronsoro’s theorem. In brief, it reduces the problem in $\mathbb{R}^{n}$ to a problem in $\mathbb{R}^{n - 1}$ via integralgeometric considerations. For the case $n = 1$, a short geometric proof already exists in the literature. A similar proof technique applies to parabolic Lipschitz functions $f \colon \mathbb{R}^{n - 1} \times \mathbb{R} \to \mathbb{R}$. A natural Dorronsoro estimate in this class is known, due to Hofmann. The method presented here allows one to reduce the parabolic problem to the Euclidean one and to obtain an elementary proof also in this setting. As a corollary, I deduce an analogue of Rademacher’s theorem for parabolic Lipschitz functions.

scholarly journals FRÉCHET INTERMEDIATE DIFFERENTIABILITY OF LIPSCHITZ FUNCTIONS ON ASPLUND SPACES

2009 ◽  
Vol 79 (2) ◽  
pp. 309-317 ◽  
Author(s):  
J. R. GILES
Keyword(s):  
Large Class ◽  
Open Subset ◽  
Lipschitz Function ◽  
Dense Subset ◽  
Lipschitz Functions ◽  
Nonempty Open Subset ◽  
Asplund Space ◽  
Asplund Spaces ◽  
Fréchet Differentiable

AbstractThe deep Preiss theorem states that a Lipschitz function on a nonempty open subset of an Asplund space is densely Fréchet differentiable. However, the simpler Fabian–Preiss lemma implies that it is Fréchet intermediately differentiable on a dense subset and that for a large class of Lipschitz functions this dense subset is residual. Results are presented for Asplund generated spaces.

BORNOLOGIES AND LOCALLY LIPSCHITZ FUNCTIONS

2014 ◽  
Vol 90 (2) ◽  
pp. 257-263 ◽  
Author(s):  
GERALD BEER ◽  
M. I. GARRIDO
Keyword(s):  
Metric Space ◽  
Lipschitz Function ◽  
Lipschitz Functions ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz Functions ◽  
The Family ◽  
Locally Lipschitz

AbstractLet$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\langle X,d \rangle $be a metric space. We characterise the family of subsets of$X$on which each locally Lipschitz function defined on$X$is bounded, as well as the family of subsets on which each member of two different subfamilies consisting of uniformly locally Lipschitz functions is bounded. It suffices in each case to consider real-valued functions.

On the Composition of Differentiable Functions

2003 ◽  
Vol 46 (4) ◽  
pp. 481-494 ◽  
Author(s):  
M. Bachir ◽  
G. Lancien
Keyword(s):  
Banach Space ◽  
Variational Principle ◽  
Linear Operator ◽  
Differentiable Function ◽  
General Result ◽  
Lipschitz Function ◽  
Lipschitz Functions ◽  
Schur Property ◽  
Differentiable Functions ◽  
Fréchet Differentiable

AbstractWe prove that a Banach space X has the Schur property if and only if every X-valued weakly differentiable function is Fréchet differentiable. We give a general result on the Fréchet differentiability of f ○ T, where f is a Lipschitz function and T is a compact linear operator. Finally we study, using in particular a smooth variational principle, the differentiability of the semi norm ‖ ‖lip on various spaces of Lipschitz functions.

scholarly journals Smooth Approximation of Lipschitz Functions on Finsler Manifolds

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
M. I. Garrido ◽  
J. A. Jaramillo ◽  
Y. C. Rangel
Keyword(s):  
Lipschitz Function ◽  
Composition Operators ◽  
Lipschitz Functions ◽  
Finsler Manifold ◽  
Positive Continuous Function ◽  
Smooth Approximation ◽  
Finsler Manifolds ◽  
Lipschitz Constants ◽  
Bounded Derivative

We study the smooth approximation of Lipschitz functions on Finsler manifolds, keeping control on the corresponding Lipschitz constants. We prove that, given a Lipschitz functionf:M→ℝdefined on a connected, second countable Finsler manifoldM, for each positive continuous functionε:M→(0,∞)and eachr>0, there exists aC1-smooth Lipschitz functiong:M→ℝsuch that|f(x)-g(x)|≤ε(x), for everyx∈M, andLip(g)≤Lip(f)+r. As a consequence, we derive a completeness criterium in the class of what we call quasi-reversible Finsler manifolds. Finally, considering the normed algebraCb1(M)of allC1functions with bounded derivative on a complete quasi-reversible Finsler manifoldM, we obtain a characterization of algebra isomorphismsT:Cb1(N)→Cb1(M)as composition operators. From this we obtain a variant of Myers-Nakai Theorem in the context of complete reversible Finsler manifolds.

scholarly journals Lipschitz functions on the infinite-dimensional torus

2016 ◽  
Vol 18 (01) ◽  
pp. 1550029
Author(s):  
Dmitry Faifman ◽  
Bo’az Klartag
Keyword(s):  
Lipschitz Function ◽  
Lipschitz Functions ◽  
Dimensional Torus ◽  
Infinite Dimensional ◽  
Number Formula ◽  
Infinite Dimensional Torus

We discuss the spectrum phenomenon for Lipschitz functions on the infinite-dimensional torus. Suppose that [Formula: see text] is a measurable, real-valued, Lipschitz function on the torus [Formula: see text]. We prove that there exists a number [Formula: see text] with the following property: For any [Formula: see text], there exists a parallel, infinite-dimensional subtorus [Formula: see text] such that the restriction of the function [Formula: see text] to the subtorus [Formula: see text] has an [Formula: see text]-norm of at most [Formula: see text].

Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous Spaces

2012 ◽  
Vol 55 (3) ◽  
pp. 646-662 ◽  
Author(s):  
Jiang Zhou ◽  
Bolin Ma
Keyword(s):  
Hardy Spaces ◽  
Lipschitz Function ◽  
Homogeneous Spaces ◽  
Lipschitz Functions ◽  
Marcinkiewicz Integral ◽  
Type Condition ◽  
Lebesgue Spaces ◽  
Lipschitz Spaces ◽  
Nondoubling Measure

AbstractUnder the assumption that μ is a nondoubling measure, we study certain commutators generated by the Lipschitz function and the Marcinkiewicz integral whose kernel satisfies a Hörmander type condition. We establish the boundedness of these commutators on the Lebesgue spaces, Lipschitz spaces, and Hardy spaces. Our results are extensions of known theorems in the doubling case.

scholarly journals Lipschitz Functions on Expanders are Typically Flat

2013 ◽  
Vol 22 (4) ◽  
pp. 566-591 ◽  
Author(s):  
RON PELED ◽  
WOJCIECH SAMOTIJ ◽  
AMIR YEHUDAYOFF
Keyword(s):  
Exponential Decay ◽  
Lipschitz Function ◽  
Lipschitz Functions ◽  
Expander Graphs ◽  
Double Exponential ◽  
Random Integer ◽  
Typical Behaviour ◽  
Good Expansion

This work studies the typical behaviour of random integer-valued Lipschitz functions on expander graphs with sufficiently good expansion. We consider two families of functions: M-Lipschitz functions (functions which change by at most M along edges) and integer-homomorphisms (functions which change by exactly 1 along edges). We prove that such functions typically exhibit very small fluctuations. For instance, we show that a uniformly chosen M-Lipschitz function takes only M+1 values on most of the graph, with a double exponential decay for the probability of taking other values.

Sign in / Sign up

Export Citation Format

Share Document