Real-valued Lipschitz Function Research Articles

Extension procedures for lattice Lipschitz operators on Euclidean spaces

We present a new class of Lipschitz operators on Euclidean lattices that we call lattice Lipschitz maps, and we prove that the associated McShane and Whitney formulas provide the same extension result that holds for the real valued case. Essentially, these maps satisfy a (vector-valued) Lipschitz inequality involving the order of the lattice, with the peculiarity that the usual Lipschitz constant becomes a positive real function. Our main result shows that, in the case of Euclidean space, being lattice Lipschitz is equivalent to having a diagonal representation, in which the coordinate coefficients are real-valued Lipschitz functions. We also show that in the linear case the extension of a diagonalizable operator from the values in their eigenvectors coincide with the operator obtained both from the McShane and the Whitney formulae. Our work on such extension/representation formulas is intended to follow current research on the design of machine learning algorithms based on the extension of Lipschitz functions.

Non-Linear Operators and Differentiability of Lipschitz Functions

In this work we provide a characterization of distinct types of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological and non-linear framework. Restricted to the linear case, we can apply our results to compact, weakly-compact, limited and completely continuous linear operators. Moreover, our results yield a characterization of Gelfand-Phillips spaces and recover some known results about Schur spaces and reflexive spaces concerning the differentiability of real-valued Lipschitz functions.

On large ℓ₁-sums of Lipschitz-free spaces and applications

We prove that the Lipschitz-free space over a Banach space X X of density κ \kappa , denoted by F ( X ) \mathcal {F}(X) , is linearly isomorphic to its ℓ 1 \ell _1 -sum ( ⨁ κ F ( X ) ) ℓ 1 \left (\bigoplus _{\kappa }\mathcal {F}(X)\right )_{\ell _1} . This provides an extension of a previous result from Kaufmann in the context of non-separable Banach spaces. Further, we obtain a complete classification of the spaces of real-valued Lipschitz functions that vanish at 0 0 over a L p \mathcal {L}_p -space. More precisely, we establish that, for every 1 ≤ p ≤ ∞ 1\leq p\leq \infty , if X X is a L p \mathcal {L}_p -space of density κ \kappa , then L i p 0 ( X ) \mathrm {Lip}_0(X) is either isomorphic to L i p 0 ( ℓ p ( κ ) ) \mathrm {Lip}_0(\ell _p(\kappa )) if p > ∞ p>\infty , or L i p 0 ( c 0 ( κ ) ) \mathrm {Lip}_0(c_0(\kappa )) if p = ∞ p=\infty .

Projecting Lipschitz Functions Onto Spaces of Polynomials

The Banach space $$\mathcal {P}({}^2X)$$ of 2-homogeneous polynomials on the Banach space X can be naturally embedded in the Banach space $${\mathrm{Lip}_0}(B_X)$$ of real-valued Lipschitz functions on $$B_X$$ that vanish at 0. We investigate whether $$\mathcal {P}({}^2X)$$ is a complemented subspace of $${\mathrm{Lip}_0}(B_X)$$ . This line of research can be considered as a polynomial counterpart to a classical result by Joram Lindenstrauss, asserting that $$\mathcal {P}({}^1X)=X^*$$ is complemented in $${\mathrm{Lip}_0}(B_X)$$ for every Banach space X. Our main result asserts that $$\mathcal {P}({}^2X)$$ is not complemented in $${\mathrm{Lip}_0}(B_X)$$ for every Banach space X with non-trivial type.

Global Lipschitz extension preserving local constants

The intent of this short note is to extend real valued Lipschitz functions on metric spaces, while locally preserving the asymptotic Lipschitz constant. We then apply this results to give a simple and direct proof of the fact that Sobolev spaces on metric measure spaces defined with a relaxation approach à la Cheeger are invariant under isomorphism class of mm-structures.

Real-valued Lipschitz functions and metric properties of functions

The purpose of this article is to explore the very general phenomenon that a function between metric spaces has a particular metric property if and only if whenever it is followed in a composition by an arbitrary real-valued Lipschitz function, the composition has this property. The key tools we use are the Efremovič lemma [21] and a theorem of Garrido and Jaramillo [22] that says that a function h between metric spaces is Lipschitz if and only if whenever it is followed by a Lipschitz real-valued function in a composition, the composition is Lipschitz. We also present a streamlined proof of the Garrido-Jaramillo result itself, but one that still relies on their natural continuous linear operator from the Lipschitz space for the target space to the Lipschitz space for the domain. Separately, we include a highly applicable uniform closure theorem that yields the most important uniform density theorems for Lipschitz-type functions as special cases.

McShane’s Extension Theorem Revisited

A fundamental extension theorem of McShane states that a bounded real-valued uniformly continuous function defined on a nonempty subset A of a metric space 〈X, d〉 can be extended to a uniformly continuous function on the entire space. In the first half of this note, we obtain McShane’s Extension Theorem from the simpler fact that a real-valued Lipschitz function defined on a nonempty subset of the space has a Lipschitz constant preserving extension to the entire space. In the second half of the note, we use this theorem to give an elementary proof of the equivalence of the most important characterizations of metric spaces in which the real-valued uniformly continuous functions form a ring. These characterizations of such a basic property, due to Cabello-Sanchez and separately Bouziad and Sukhacheva, are remarkably recent.

Cone unrectifiable sets and non-differentiability of Lipschitz functions

We provide sufficient conditions for a set E ⊂ ℝn to be a non-universal differentiability set, i.e., to be contained in the set of points of non-differentiability of a real-valued Lipschitz function. These conditions are motivated by a description of the ideal generated by sets of non-differentiability of Lipschitz self-maps of ℝn given by Alberti, Csornyei and Preiss, which eventually led to the result of Jones and Csornyei that for every Lebesgue null set E in ℝn there is a Lipschitz map f: ℝn → ℝn not differentiable at any point of E, even though for n > 1 and for Lipschitz functions from ℝn to ℝ there exist Lebesgue null universal differentiability sets. Among other results, we show that the new class of Lebesgue null sets introduced here contains all uniformly purely unrectifiable sets and gives a quantified version of the result about non-differentiability in directions outside the decomposability bundle with respect to a Radon measure.

On the Dirichlet problem on Lorentz and Orlicz spaces with applications to Schwarz–Christoffel domains

It is known (see [14]) that, for every Lipschitz domain Ω on the planeΩ={x+iy:y>ν(x)}, with ν a real valued Lipschitz function, there exists 1≤p0<2 so that the Dirichlet problem has a solution for every function f∈Lp(ds) and every p∈(p0,∞). Moreover, if p0>1, the result is false for every p≤p0. The purpose of this paper is to study in more detail what happens at the endpoint p0; that is, we want to find spaces X⊂Lp0 so that the Dirichlet problem is solvable for every f∈X. These spaces X will be either the Lorentz space Lp0,1(ds) or some type of logarithmic Orlicz space. Our results will be applied to the special case of Schwarz–Christoffel Lipschitz domains, among others, for which we explicitly compute the value of p0.

A Measure Zero UDS in the Heisenberg Group

We show that the Heisenberg group contains a measure zero set N such that every real-valued Lipschitz function is Pansu differentiable at a point of N.

Two classes of metric spaces

&lt;p&gt;The class of metric spaces (X,d) known as small-determined spaces, introduced by Garrido and Jaramillo, are properly defined by means of some type of real-valued Lipschitz functions on X. On the other hand, B-simple metric spaces introduced by Hejcman are defined in terms of some kind of bornologies of bounded subsets of X. In this note we present a common framework where both classes of metric spaces can be studied which allows us to see not only the relationships between them but also to obtain new internal characterizations of these metric properties.&lt;/p&gt;

A note on a Residual Subset of Lipschitz Functions on Metric Spaces

AbstractLet (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of real-valued Lipschitz functions with non-zero pointwise Lipschitz constant m-almost everywhere is residual, and hence dense, in the Banach space of Lipschitz and bounded functions. The result is the metric analogous to a result proved for real-valued Lipschitz maps defined on ℝ2 by Alberti et al.

A computational model for multi-variable differential calculus

We develop a domain-theoretic computational model for multi-variable differential calculus, which for the first time gives rise to data types for piecewise differentiable or more generally Lipschitz functions, by constructing an effectively given continuous Scott domain for real-valued Lipschitz functions on finite dimensional Euclidean spaces. The model for real-valued Lipschitz functions of n variables is built as a sub-domain of the product of two domains by tupling together consistent information about locally Lipschitz functions and their differential properties as given by their L-derivative or equivalently Clarke gradient, which has values given by non-empty, convex and compact subsets of Rn. To obtain a computationally practical framework, the derivative information is approximated by the best fit compact hyper-rectangles in Rn. In this case, we show that consistency of the function and derivative information can be decided by reducing it to a linear programming problem. This provides an algorithm to check consistency on the rational basis elements of the domain, implying that the domain can be equipped with an effective structure and giving a computable framework for multi-variable differential calculus. We also develop a domain-theoretic, interval-valued, notion of line integral and show that if a Scott continuous function, representing a non-empty, convex and compact valued vector field, is integrable, then its interval-valued integral over any closed piecewise C1 path contains zero. In the case that the derivative information is given in terms of compact hyper-rectangles, we use techniques from the theory of minimal surfaces to deduce the converse result: a hyper-rectangular valued vector field is integrable if its interval-valued line integral over any piecewise C1 path contains zero. This gives a domain-theoretic extension of the fundamental theorem of path integration. Finally, we construct the least and the greatest piecewise linear functions obtained from a pair of function and hyper-rectangular derivative information. When the pair is consistent, this provides the least and greatest maps to witness consistency.

On the Quasi-linear Elliptic PDE $${-\nabla \cdot ( \nabla{u}/ \sqrt{1-| \nabla{u} |^2}) = 4 \pi \sum_k a_k \delta_{s_k}}$$ in Physics and Geometry

It is shown that for each finite number of Dirac measures supported at points $s_n$ in three-dimensional Euclidean space, with given amplitudes $a_n$, there exists a unique real-valued Lipschitz function $u$, vanishing at infinity, which distributionally solves the quasi-linear elliptic partial differential equation of divergence form $-\nabla\cdot(\nabla{u}/\sqrt{1-|\nabla{u}|^2})=4\pi\sum_{n=1}^N a_n \delta_{s_n}$. Moreover, $u$ is real analytic away from the $s_n$. The result can be interpreted in at least two ways: (a) for any number of point charges of arbitrary magnitude and sign at prescribed locations $s_n$ in three-dimensional Euclidean space there exists a unique electrostatic field which satisfies the Maxwell-Born-Infeld field equations smoothly away from the point charges and vanishes as $|s|\to\infty$; (b) for any number of integral mean curvatures assigned to locations $s_n$ there exists a unique asymptotically flat, almost everywhere space-like maximal slice with point defects of Minkowski spacetime, having lightcone singularities over the $s_n$ but being smooth otherwise, and whose height function vanishes as $|s|\to\infty$. No struts between the point singularities ever occur.

A compact universal differentiability set with Hausdorff dimension one

We give a short proof that any non-zero Euclidean space has a compact subset of Hausdorff dimension one that contains a differentiability point of every real-valued Lipschitz function defined on the space.

Smooth extension of functions on a certain class of non-separable Banach spaces

Let us consider a Banach space X with the property that every real-valued Lipschitz function f can be uniformly approximated by a Lipschitz, C 1 -smooth function g with Lip ( g ) ⩽ C Lip ( f ) (with C depending only on the space X). This is the case for a Banach space X bi-Lipschitz homeomorphic to a subset of c 0 ( Γ ) , for some set Γ, such that the coordinate functions of the homeomorphism are C 1 -smooth (Hájek and Johanis, 2010 [10]). Then, we prove that for every closed subspace Y ⊂ X and every C 1 -smooth (Lipschitz) function f : Y → R , there is a C 1 -smooth (Lipschitz, respectively) extension of f to X. We also study C 1 -smooth extensions of real-valued functions defined on closed subsets of X. These results extend those given in Azagra et al. (2010) [4] to the class of non-separable Banach spaces satisfying the above property.

A compact null set containing a differentiability point of every Lipschitz function

We prove that in a Euclidean space of dimension at least two, there exists a compact set of Lebesgue measure zero such that any real-valued Lipschitz function defined on the space is differentiable at some point in the set. Such a set is constructed explicitly.

Fréchet differentiability of Lipschitz functions via a variational principle

We prove a new variational principle which in particular does not assume the completeness of the domain. As an application we give a new, more natural, proof of the fact that a real valued Lipschitz function on an Asplund space has points of Fréchet differentiability.

Flattening functions on flowers

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.

Lipschitz functions with unexpectedly large sets of nondifferentiability points

It is known that every Gδ subset E of the plane containing a dense set of lines, even if it has measure zero, has the property that every real‐valued Lipschitz function on ℝ2 has a point of differentiability in E. Here we show that the set of points of differentiability of Lipschitz functions inside such sets may be surprisingly tiny: we construct a Gδ set E ⊂ ℝ2 containing a dense set of lines for which there is a pair of real‐valued Lipschitz functions on ℝ2 having no common point of differentiability in E, and there is a real‐valued Lipschitz function on ℝ2 whose set of points of differentiability in E is uniformly purely unrectifiable.