Fourier transforms and generalized Lipschitz classes in uniform metric

For functions f∈L¹(R)∩C(R) with Fourier transforms in L¹(R) we give necessary and sufficient conditions for f to belong to the generalized Lipschitz classes Hω,m and hω,m in terms of behavior of .     

About the Lipschitz property of the metric projection in the Hilbert space

We consider the metric projection operator from the real Hilbert space onto a strongly convex set. We prove that the restriction of this operator on the complement of some neighborhood of the strongly convex set is Lipschitz continuous with the Lipschitz constant strictly less than 1. This property characterizes the class of strongly convex sets and (to a certain degree) the Hilbert space. We apply the results obtained to the question concerning the rate of convergence for the gradient projection algorithm with differentiable convex function and strongly convex set.     

Fourier transforms in generalized Lipschitz classes

We obtain sufficient conditions for the Fourier transform of a function f ∈ L ₁(?) to belong to generalized Lipschitz classes defined by the modulus of smoothness of order m. The sharpness of these conditions is established in the cases when f(t) ≥ 0 on ? or tf (t) ≥ 0 on ?.     

Lipschitz metric for the periodic Camassa-Holm equation

We study the stability of conservative solutions of the Cauchy problem for the Camassa-Holm equation ut−uxxt+κux+3uux−2uxuxx−uuxxx=0 with periodic initial data u₀. In particular, we derive a new Lipschitz metric dD with the property that for two solutions u and v of the equation we have dD(u(t),v(t))?eCtdD(u₀,v₀). The relationship between this metric and usual norms in and is clarified.     

Quasiminima of the Lipschitz extension problem

In this paper, we extend the notion of quasiminimum to the framework of supremum functionals by studying the model case

Pointwise Lipschitz functions on metric spaces

For a metric space X, we study the space D^∞(X) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D^∞(X) is compared with the space LIP^∞(X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D^∞(X) with the Newtonian-Sobolev space N1,∞(X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D^∞(X)=N1,∞(X).     

Semi-Infinite Optimization under Convex Function Perturbations: Lipschitz Stability

This paper is devoted to the study of the stability of the solution map for the parametric convex semi-infinite optimization problem under convex function perturbations in short, PCSI. We establish sufficient conditions for the pseudo-Lipschitz property of the solution map of PCSI under perturbations of both objective function and constraint set. The main result obtained is new even when the problems under consideration reduce to linear semi-infinite optimization. Examples are given to illustrate the obtained results.     

Lipschitz classes and the Hardy-Littlewood property

We study the geometry of plane domains and the uniform Hölder continuity properties of analytic functions.With 1 Figure     

Markov chains,Riesz transforms and Lipschitz maps

it is shown that a version of Maurey's extension theorem holds for Lipschitz maps between metric spaces satisfying certain geometric conditions, analogous to type and cotype. As a consequence, a classical Theorem of Kirszbraun can be generalised to include maps intoL _p , 1<p<2. These conditions describe the wandering of symmetric Markov processes in the spaces in question. Estimates are obtained for the root-mean-square wandering of such processes in theL _p spaces. The duality theory for these geometric conditions (in normed spaces) is shown to be closely related to the behavior of the Riesz transforms associated to Markov chains. Several natural open problems are collected in the final chapter.Supported in part by NSF DMS-8807243.     

Lipschitz and path isometric embeddings of metric spaces

We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash C ¹ Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also show that any metric space of finite Hausdorff dimension can be embedded in some Euclidean space via a Lipschitz map.     

On Lipschitz extension from finite subsets

We prove that for every n ∈ ? there exists a metric space (X, d _X), an n-point subset S ? X, a Banach space (Z, \({\left\| \right\|_Z}\)) and a 1-Lipschitz function f: S → Z such that the Lipschitz constant of every function F: X → Z that extends f is at least a constant multiple of \(\sqrt {\log n} \). This improves a bound of Johnson and Lindenstrauss [JL84]. We also obtain the following quantitative counterpart to a classical extension theorem of Minty [Min70]. For every α ∈ (1/2, 1] and n ∈ ? there exists a metric space (X, d _X), an n-point subset S ? X and a function f: S → ?₂ that is α-Hölder with constant 1, yet the α-Hölder constant of any F: X → ?₂ that extends f satisfies \({\left\| F \right\|_{Lip\left( \alpha \right)}} > {\left( {\log n} \right)^{\frac{{2\alpha - 1}}{{4\alpha }}}} + {\left( {\frac{{\log n}}{{\log \log n}}} \right)^{{\alpha ^2} - \frac{1}{2}}}\). We formulate a conjecture whose positive solution would strengthen Ball’s nonlinear Maurey extension theorem [Bal92], serving as a far-reaching nonlinear version of a theorem of König, Retherford and Tomczak-Jaegermann [KRTJ80]. We explain how this conjecture would imply as special cases answers to longstanding open questions of Johnson and Lindenstrauss [JL84] and Kalton [Kal04].     

Operators with the Lipschitz bounded approximation property

We show that if a bounded linear operator can be approximated by a net(or sequence) of uniformly bounded finite rank Lipschitz mappings pointwisely, then it can be approximated by a net(or sequence) of uniformly bounded finite rank linear operators under the strong operator topology. As an application, we deduce that a Banach space has an(unconditional) Lipschitz frame if and only if it has an(unconditional)Schauder frame. Another immediate consequence of our result recovers the famous Godefroy-...     

Operator norm localization property of metric spaces under finite decomposition complexity

The notions of operator norm localization property and finite decomposition complexity were recently introduced in metric geometry to study the coarse Novikov conjecture and the stable Borel conjecture. In this paper we show that a metric space X has weak finite decomposition complexity with respect to the operator norm localization property if and only if X itself has the operator norm localization property. It follows that any metric space with finite decomposition complexity has the operator norm localization property. In particular, we obtain an alternative way to prove a very recent result by E. Guentner, R. Tessera and G. Yu that all countable linear groups have the operator norm localization property.     

Absolutely minimal Lipschitz extension of tree-valued mappings

We prove that every Lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal Lipschitz extension (AMLE). We relate these extensions to a stochastic game called Politics??a generalization of a game called Tug of War that has been used in Peres et?al. (J Am Math Soc 22(1):167?C210, 2009) to study real-valued AMLEs.