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David Preiss - One of the best experts on this subject based on the ideXlab platform.
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Differentiability of Lipschitz Functions in Lebesgue null sets
Inventiones mathematicae, 2014Co-Authors: David Preiss, Gareth SpeightAbstract:We show that if \(n>1\) then there exists a Lebesgue null set in \({\mathbb {R}}^{n}\) containing a point of differentiability of each Lipschitz Function \(f:{\mathbb {R}}^{n} \rightarrow {\mathbb {R}}^{n-1}\); in combination with the work of others, this completes the investigation of when the classical Rademacher theorem admits a converse. Avoidance of \(\sigma \)-porous sets, arising as irregular points of Lipschitz Functions, plays a key role in the proof.
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Lipschitz Functions with unexpectedly large sets of nondifferentiability points
Abstract and Applied Analysis, 2005Co-Authors: Marianna Csörnyei, David Preiss, Jaroslav TišerAbstract:It is known that every G δ 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 ℝ 2 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 G δ set E ⊂ ℝ 2 containing a dense set of lines for which there is a pair of real-valued Lipschitz Functions on ℝ 2 having no common point of differentiability in E , and there is a real-valued Lipschitz Function on ℝ 2 whose set of points of differentiability in E is uniformly purely unrectifiable.
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A new proof of Fréchet differentiability of Lipschitz Functions
Journal of the European Mathematical Society, 2000Co-Authors: Joram Lindenstrauss, David PreissAbstract:We give a relatively simple (self-contained) proof that every real-valued Lipschitz Function on l2 (or more generally on an Asplund space) has points of Frechet differentiability. Somewhat more generally, we show that a real-valued Lipschitz Function on a separable Banach space has points of Frechet differentiability provided that the w* closure of the set of its points of Gâteaux differentiability is norm separable.
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Affine Approximation of Lipschitz Functions and Nonlinear Quotients
Geometric And Functional Analysis, 1999Co-Authors: S. Bates, David Preiss, Joram Lindenstrauss, William B. Johnson, Gideon SchechtmanAbstract:New concepts related to approximating a Lipschitz Function between Banach spaces by affine Functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X, Y for which any Lipschitz Function from X to Y can be so approximated is obtained. This is applied to the study of Lipschitz and uniform quotient mappings between Banach spaces. It is proved, in particular, that any Banach space which is a uniform quotient of L p , 1 < p < \( \infty \), is already isomorphic to a linear quotient of L p .
A. B. Aleksandrov - One of the best experts on this subject based on the ideXlab platform.
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Interpolation by the Derivatives of Operator Lipschitz Functions
50 Years with Hardy Spaces, 2018Co-Authors: A. B. AleksandrovAbstract:Let Λ be a discrete subset of the real line ℝ. We prove that for every bounded Function φ on Λ there exists an operator Lipschitz Function f on ℝ such that f’ (t) = φ(t) for all t∈Λ. The same is true for the set of operator Lipschitz Functions f on ℝ such that f’ coincides with the non-tangential boundary values of a bounded holomorphic Function on the upper half-plane. In other words, for every bounded Function φ on Λ there exists a commutator Lipschitz Function f on the closed upper half-plane such that f’ (t) = φ(t) for all t∈Λ. The same is also true for some non-discrete countable sets Λ. Furthermore, we consider the case where Λ is a subset of the closed upper half-plane, Λ⊄ ℝ. Similar questions for commutator Lipschitz Functions on a closed subset F of ℂ are also considered.
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Operator Lipschitz Functions in Several Variables and Möbius Transformations
Journal of Mathematical Sciences, 2015Co-Authors: A. B. AleksandrovAbstract:It is proved that if f is an operator Lipschitz Function defined on ℝ n , then the Function $$ \frac{f\circ \varphi }{\left\Vert {\varphi}^{\prime}\right\Vert } $$ is also operator Lipschitz for every Mobius transformation φ with f(φ(∞)) = 0. Here ‖φ′‖ denotes the operator norm of the Jacobian matrix φ′ Similar statements for operator Lipschitz Functions defined on closed subsets of ℝ n are also obtained. Bibliography: 10 titles.
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Operator Lipschitz Functions and linear fractional transformations
Journal of Mathematical Sciences, 2013Co-Authors: A. B. AleksandrovAbstract:It is known that the Function $t^2\sin\frac1t$ is an operator Lipschitz Function on the real line ${\mathbb R}$ . We prove that the Function sin can be replaced by any operator Lipschitz Function f with f(0) = 0. In other words, for every operator Lipschitz Function f, the Function $t^2 f(\frac1t)$ is also operator Lipschitz if f(0) = 0. The Function f can be defined on an arbitrary closed subset of the complex plane ${\mathbb C}$ . Moreover, the linear fractional transformation $\frac1t$ can be replaced by every linear fractional transformation ϕ. In this case, we assert that the Function $\dfrac{f\circ\varphi}{\varphi^{\,\prime}}$ is operator Lipschitz for every operator Lipschitz Function f provided that f(ϕ( ∞ )) = 0. Bibliography: 12 titles.
Marc Lassonde - One of the best experts on this subject based on the ideXlab platform.
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Subdifferential characterization of approximate convexity: the lower semicontinuous case
Mathematical Programming B, 2009Co-Authors: Aris Daniilidis, Florence Jules, Marc LassondeAbstract:It is known that a locally Lipschitz Function f is approximately convex if, and only if, its Clarke subdifferential ∂C f is a submonotone operator. The main object of this work is to extend the above characterization to the class of lower semicontinuous Functions. To this end, we establish a new approximate mean value inequality involving three points. We also show that an analogue of the Rockafellar maximal monotonicity theorem holds for this class of Functions and we discuss the case of arbitrary subdifferentials
Yenny C. Rangel - One of the best experts on this subject based on the ideXlab platform.
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Smooth Approximation of Lipschitz Functions on Finsler Manifolds
Journal of Function Spaces and Applications, 2013Co-Authors: M. I. Garrido, Jesús A. Jaramillo, Yenny C. RangelAbstract:We study the smooth approximation of Lipschitz Functions on Finsler manifolds, keeping control on the corresponding Lipschitz constants. We prove that, given a Lipschitz Function defined on a connected, second countable Finsler manifold , for each positive continuous Function and each , there exists a -smooth Lipschitz Function such that , for every , and . As a consequence, we derive a completeness criterium in the class of what we call quasi-reversible Finsler manifolds. Finally, considering the normed algebra of all Functions with bounded derivative on a complete quasi-reversible Finsler manifold , we obtain a characterization of algebra isomorphisms as composition operators. From this we obtain a variant of Myers-Nakai Theorem in the context of complete reversible Finsler manifolds.
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Smooth approximation of Lipschitz Functions on Riemannian manifolds
Journal of Mathematical Analysis and Applications, 2007Co-Authors: Daniel Azagra, Juan Ferrera, Fernando López-mesas, Yenny C. RangelAbstract:Abstract We show that for every Lipschitz Function f defined on a separable Riemannian manifold M (possibly of infinite dimension), for every continuous e : M → ( 0 , + ∞ ) , and for every positive number r > 0 , there exists a C ∞ smooth Lipschitz Function g : M → R such that | f ( p ) − g ( p ) | ⩽ e ( p ) for every p ∈ M and Lip ( g ) ⩽ Lip ( f ) + r . Consequently, every separable Riemannian manifold is uniformly bumpable. We also present some applications of this result, such as a general version for separable Riemannian manifolds of Deville–Godefroy–Zizler's smooth variational principle.
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Smooth Approximation of Lipschitz Functions on Riemannian manifolds
arXiv: Differential Geometry, 2006Co-Authors: Daniel Azagra, Juan Ferrera, Fernando López-mesas, Yenny C. RangelAbstract:We show that for every Lipschitz Function $f$ defined on a separable Riemannian manifold $M$ (possibly of infinite dimension), for every continuous $\epsilon:M\to (0,+\infty)$, and for every positive number $r>0$, there exists a $C^\infty$ smooth Lipschitz Function $g:M\to\mathbb{R}$ such that $|f(p)-g(p)|\leq\epsilon(p)$ for every $p\in M$ and $\textrm{Lip}(g)\leq\textrm{Lip}(f)+r$. Consequently, every separable Riemannian manifold is uniformly bumpable. We also present some applications of this result, such as a general version for separable Riemannian manifolds of Deville-Godefroy-Zizler's smooth variational principle.
Aris Daniilidis - One of the best experts on this subject based on the ideXlab platform.
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Characterization of Filippov representable maps and Clarke subdifferentials
2020Co-Authors: Mira Bivas, Aris Daniilidis, Marc QuincampoixAbstract:The ordinary differential equationẋ(t) = f (x(t)), t ≥ 0, for f measurable, is not sufficiently regular to guarantee existence of solutions. To remedy this we may relax the problem by replacing the Function f with its Filippov regularization F f and consider the differential inclusioṅ x(t) ∈ F f (x(t)) which always has a solution. It is interesting to know, inversely, when a set-valued map Φ can be obtained as the Filippov regularization of a (single-valued, measurable) Function. In this work we give a full characterization of such set-valued maps, hereby called Filippov representable. This characterization also yields an elegant description of those maps that are Clarke subdifferentials of a Lipschitz Function.
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Subdifferential characterization of approximate convexity: the lower semicontinuous case
Mathematical Programming B, 2009Co-Authors: Aris Daniilidis, Florence Jules, Marc LassondeAbstract:It is known that a locally Lipschitz Function f is approximately convex if, and only if, its Clarke subdifferential ∂C f is a submonotone operator. The main object of this work is to extend the above characterization to the class of lower semicontinuous Functions. To this end, we establish a new approximate mean value inequality involving three points. We also show that an analogue of the Rockafellar maximal monotonicity theorem holds for this class of Functions and we discuss the case of arbitrary subdifferentials