Compact polynomials between Banach spaces

Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEDGYCITpu

Variations on the Banach-Stone theorem

This paper is based on a series of lectures delivered during a 2001 summer course of the University of Cantabria in Spain. The central theme is the characterization of a topological space X in terms of the topological-algebraic structure of suitably chosen subspaces of the space C(X) of continuous functions on X . A huge variety of corresponding results is presented. After having discussed the classical Banach-Stone theorem, the authors present several more recent results which characterize a locally compact space X through the isometric/isomorphic structure of particular subspaces of C 0 (X) , the space of continuous functions on X which vanish at infinity. Another type of results allows to recover a complete metric space from spaces of bounded uniformly continuous functions, or of Lipschitz continuous functions, which take values in particular Banach spaces. Moreover, the paper contains a number of results regarding the characterization of a space X through algebraic properties of appropriate subspaces of C(X) , e.g., certain subalgebras. The problem of when isomorphy of spaces of differentiable functions on Banach manifolds entails isomorphy of the underlying manifolds is discussed in detail. The final chapter is devoted to the problem to decide when, for complete metric spaces X and Y , the existence of an isomorphism between suitable lattices of functions of uniformly continuous functions on X and Y , respectively, entails that X and Y are uniformly homeomorphic (similar for Lipschitz maps)

Extension of bilinear forms from subespaces of L1 -space

We study the extension of bilinear forms from a given subspace of an L1 -space to the whole space. Precisely, an isomorphic embedding j: E → X is said to be (linearly) 2-exact if bilinear forms on E can be (linear and continuously) extended to X through j . We present some necesary and some sufficient conditions for an embedding j: E → X to be 2-exact when X is an L1 -space

Asymptotic Smoothness, Convex Envelopes and Polynomial Norms

We introduce a suitable notion of asymptotic smoothness on infinite dimensional Banach spaces, and we prove that, under some structural restrictions on the space, the convex envelope of an asymptotically smooth function is asymptotically smooth. Furthermore, we study convexity and smoothness properties of polynomial norms, and we obtain that a polynomial norm of degree N has modulus of convexity of power type N

Geometric characterizations of p-Poincaré inequalities in the metric setting

We prove that a locally complete metric space endowed with a doubling measure satisfies an infinity-Poincare inequality if and only if given a null set, every two points can be joined by a quasiconvex curve which "almost avoids" that set. As an application, we characterize doubling measures on R satisfying an infinity-Poincare inequality. For Ahlfors Q-regular spaces, we obtain a characterization of p-Poincare inequality for p > Q in terms of the p-modulus of quasiconvex curves connecting pairs of points in the space. A related characterization is given for the case Q - 1 < p <= Q

Almost classical solutions of Hamilton-Jacobi equations

We study the existence of everywhere differentiable functions which are almost everywhere solutions of quite general Hamilton-Jacobi equations on open subsets of R(d) or on d-dimensional manifolds whenever d >= 2. In particular, when M is a Riemannian manifold, we prove the existence of a differentiable function a on M which satisfies the Eikonal equation parallel to del u(x)parallel to(x) = 1 almost everywhere on M

Global inversion of nonsmooth mappings using pseudo-Jacobian matrices

We study the global inversion of a continuous nonsmooth mapping f : R-n -> R-n, which may be non-locally Lipschitz. To this end, we use the notion of pseudo-Jacobian map associated to f, introduced by Jeyakumar and Luc, and we consider a related index of regularity for f. We obtain a characterization of global inversion in terms of its index of regularity. Furthermore, we prove that the Hadamard integral condition has a natural counterpart in this setting, providing a sufficient condition for global invertibility.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEMINECO Project (Spain)CONICYT-Chile through FONDECYT projectpu

Surjection and inversion for locally Lipschitz maps between Banach spaces

We study the global invertibility of non-smooth, locally Lipschitz maps between infinite-dimensional Banach spaces, using a kind of Palais-Smale condition. To this end, we consider the Chang version of the weighted Palais-Smale condition for locally Lipschitz functionals in terms of the Clarke subdifferential, as well as the notion of pseudo-Jacobians in the infinite-dimensional setting, which are the analog of the pseudo-Jacobian matrices defined by Jeyakumar and Luc. Using these notions, we derive our results about existence and uniqueness of solution for nonlinear equations. In particular, we give a version of the classical Hadamard integral condition for global invertibility in this context

Global inversion of nonsmooth mappings on Finsler manifolds

We consider the problem of finding sufficient conditions for a locally Lipschitz mapping between Finsler manifolds to be a global homeomorphism. For this purpose, we develop the notion of Clarke generalized differential in this context and, using this, we obtain a version of the Hadamard integral condition for invertibility. As consequences, we deduce some global inversion and global injectivity results for Lipschitz mappings on R-n in terms of spectral conditions of its Clarke generalized differential

Separating polynomials on Banach spaces

In this paper we survey some recent results concerning separating polynomials on real Banach spaces. By this we mean a polynomial which separates the origin from the unit sphere of the space, thus providing an analog of the separating quadratic form on Hilbert space