Surjective isometries on spaces of vector valued continuous and Lipschitz functions

For X a compact Hausdorff topological space and F a strictly convex and reflexive Banach space we characterize the surjective isometries of certain subspaces of C(X, F). It follows from this characterization that surjective isometries on spaces of vector valued Lipschitz functions equipped with a p-norm are also weighted composition operators.     

A note on isometries of Lipschitz spaces

The main purpose of this article is to generalize a recent result about isometries of Lipschitz spaces. Botelho, Fleming and Jamison [2] described surjective linear isometries between vector-valued Lipschitz spaces under certain conditions. In this article, we extend the main result of [2] by removing the quasi-sub-reflexivity condition from Banach spaces.     

Projections and averages of isometries on Lipschitz spaces

We characterize projections on spaces of Lipschitz functions expressed as the average of two and three linear surjective isometries. Generalized bi-circular projections are the only projections on these spaces given as the convex combination of two surjective isometries.     

Extreme points and isometries on vector-valued Lipschitz spaces

For a Banach space E and a compact metric space (X,d), a function F:X→E is a Lipschitz function if there exists k>0 such that

     

Noncompactness and noncompleteness in isometries of Lipschitz spaces

We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions Lip(X,E) and Lip(Y,F), for strictly convex normed spaces E and F and metric spaces X and Y:

(i)

Characterize those base spaces X and Y for which all isometries are weighted composition maps.

(ii)

Give a condition independent of base spaces under which all isometries are weighted composition maps.

(iii)

Provide the general form of an isometry, both when it is a weighted composition map and when it is not.

In particular, we prove that requirements of completeness on X and Y are not necessary when E and F are not complete, which is in sharp contrast with results known in the scalar context.     

Local mappings on spaces of differentiable functions

We study conditions under which a functionalF(u, B) defined for everyu∈C ^k(Ω;R ^m ) and every Borel subsetB of Ω admits the integral representation

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).     

Cube formulas on spaces of continuously differentiable functions

Siberian Mathematical Journal -     

Isometries and almost isometries between spaces of continuous functions

We characterize the isometries fromC(X) intoC(Y) whereX andY are compact metric spaces. We give necessary and sufficient conditions on an isometry from a subset ofC(X) intoC(Y) to have an extension to the whole space. It is also shown that an almost isometry from the unit ball ofC(X) into the unit ball ofC(Y) is near to an isometry.     

Isometries on spaces of vector valued Lipschitz functions

This paper gives a characterization of a class of surjective isometries on spaces of Lipschitz functions with values in a finite dimensional complex Hilbert space.