Bilipschitz mappings and strong weights.
On the nonexistence of bilipschitz parametrizations and geometric problems about A-weights.
How can one recognize when a metric space is bilipschitz equivalent to an Euclidean space? One should not take the abstraction of metric spaces too seriously here; subsets of R are already quite interesting. It is easy to generate geometric conditions which are necessary for bilipschitz equivalence, but it is not clear that such conditions should ever be sufficient. The main point of this paper is that the optimistic conjectures about the existence of bilipschitz parametrizations are wrong. In other...
Interpolation of Banach spaces, differential geometry and differential equations.
In recent years the study of interpolation of Banach spaces has seen some unexpected interactions with other fields. (...) In this paper I shall discuss some more interactions of interpolation theory with the rest of mathematics, beginning with some joint work with Coifman [CS]. Our basic idea was to look for the methods of interpolation that had interesting PDE's arising as examples.
Measure-preserving quality within mappings.
In [6], Guy David introduced some methods for finding controlled behavior in Lipschitz mappings with substantial images (in terms of measure). Under suitable conditions, David produces subsets on which the given mapping is bilipschitz, with uniform bounds for the bilipschitz constant and the size of the subset. This has applications for boundedness of singular integral operators and uniform rectifiability of sets, as in [6], [7], [11], [13]. Some special cases of David's results, concerning projections...
Good metric spaces without good parameterizations.
A classical problem in geometric topology is to recognize when a topological space is a topological manifold. This paper addresses the question of when a metric space admits a quasisymmetric parametrization by providing examples of spaces with many Eucledian-like properties which are nonetheless substantially different from Euclidean geometry. These examples are geometrically self-similar versions of classical topologically self-similar examples from geometric topology, and they can be realized...
Quasisymmetry, measure and a question of Heinonen.
In this paper we resolve in the affirmative a question of Heinonen on the absolute continuity of quasisymmetric mappings defined on subsets of Euclidean spaces. The main ingredients in the proof are extension results for quasisymmetric mappings and metric doubling measures.
Happy fractals and some aspects of analysis on metric spaces.
There has been a lot of interest and activity along the general lines of analysis on metric spaces recently, as in [2], [3], [26], [40], [41], [46], [48], [49], [51], [82], [83], [89], for instance. Of course this is closely related to and involves ideas concerning spaces of homogeneous type, as in [18], [19], [66], [67], [92], as well as sub-Riemannian spaces, e.g., [8], [9], [34], [47], [52], [53], [54], [55], [68], [70], [72], [73], [84], [86], [88]. In the present survey we try to give an introduction...
Some remarks about metric spaces, spherical mappings, functions and their derivatives.
If p ∈ R, then we have the radial projection map from R {p} onto a sphere. Sometimes one can construct similar mappings on metric spaces even when the space is nontrivially different from Euclidean space, so that the existence of such a mapping becomes a sign of approximately Euclidean geometry. The existence of such spherical mappings can be used to derive estimates for the values of a function in terms of its gradient, which can then be used to derive Sobolev inequalities, etc. In this paper...
Some topics concerning homeomorphic parameterizations.
In this survey, we consider several questions pertaining to homeomorphisms, including criteria for their existence in certain circumstances, and obstructions to their existence.
Real analysis, quantitative topology, and geometric complexity.
In this paper, we give an overview of some topics involving behavior of homeomorphisms and ways in which real analysis can arise in geometric settings.
Bilipschitz embeddings of metric spaces into euclidean spaces.
When does a metric space admit a bilipschitz embedding into some finite-dimensional Euclidean space? There does not seem to be a simple answer to this question. Results of Assouad [A1], [A2], [A3] do provide a simple answer if one permits some small ("snowflake") deformations of the metric, but unfortunately these deformations immediately disrupt some basic aspects of geometry and analysis, like rectifiability, differentiability, and curves of finite length. Here we discuss a (somewhat technical)...
Surfaces quasiminimales de codimension 1 : un morceau de démonstration
Uniform rectifiability and singular sets
Harmonic analysis and the geometry of subsets of R.
This subject has several natural points of view, but we shall start with the one that corresponds to the following question: Is there something like Littlewood-Paley theory which is useful for analyzing the geometry of subsets of R, in much the same way that traditional Littlewood-Paley theory is good for analyzing functions and operators?
Regular mappings between dimensions
The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by regular mappings (reviewed in Section 1), in which some non-bilipschitz behavior is allowed, but with limitations on this, and in a quantitative way. In this paper we look at a class of mappings called (s, t)-. These mappings are the same as ordinary regular mappings when s = t, but otherwise they behave somewhat like projections....
Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation.
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