Introduction What are Poincare and Sobolev inequalities? Poincare inequalities, pointwise estimates, and Sobolev classes Examples and necessary conditions Sobolev type inequalities by means of Riesz potentials Trudinger inequality A version of the Sobolev embedding theorem on spheres Rellich-Kondrachov Sobolev classes in John domains Poincare inequality: examples Carnot-Caratheodory spaces Graphs Applications to PDE and nonlinear potential theory Appendix References.

Sobolev met Poincaré

This paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry. The principal message is that such a theory is both relevant and viable. The first main issue is the problem of definition, which we next describe. Quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount. This requirement makes sense in every metric space. Given a homeomorphism f from a metric space X to a metric space Y , then for x∈X and r>0 set

/pdf/quasiconformal-maps-in-metric-spaces-with-controlled-24nmi0p2hm.pdf

Quasiconformal maps in metric spaces with controlled geometry

The yield of alkenols and cycloalkenols is substantially improved by carrying out the reaction of olefins with formaldehyde in the presence of selected catalysts. In accordance with one embodiment, alk-3-en-1-ols are produced in good yields from isobutylene and formaldehyde in the presence of organic carboxylic acid salts of Group IB metals.

The Organization of the Cochlear Receptor

Let V be a smooth manifold where we distinguish a subset H in the set of all piecewise smooth curves c in V. We assume that H is defined by a local condition on curves, i.e. if c is divided into segments c 1,…, c k , then 

$$ c\, \in \,\mathcal{H} \Leftrightarrow \,{c_i} \in \mathcal{H},\,i = 1, \ldots ,k. $$

/pdf/carnot-caratheodory-spaces-seen-from-within-3y7qhthcfm.pdf

Carnot-Carathéodory spaces seen from within

After Hormander's fundamental paper on hypoellipticity [54], the study of partial differential equations arising from families of noncommuting vector fields has developed significantly. In this paper we study some basic functional and geometric properties of general families of vector fields that include the Hormander type as a special case. Similar to their classical counterparts, such properties play an important role in the analysis of the relevant differential operators (both linear and nonlinear). To motivate our results, we recall some classical inequalities. Let E C R" be a Caccioppoli set (a measurable set having a locally finite perimeter); then one has the isoperimetric inequality

Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces

Foundations for the Theory of Quasiconformal Mappings on the Heisenberg Group

On etudie les applications conformes et quasi conformes sur le groupe de Heisenberg. La metrique invariante a gauche naturelle sur cet espace est non isotrope sous des dilatations et ne peut se decrire par une metrique de Riemann. Ceci engendre de nouveaux phenomenes

Quasiconformal mappings on the Heisenberg group

Functions of bounded mean oscillation and quasiconformal mappings

Ordinary Differential Equations and Quasiconformal Mappings.

On H–type groups N the left invariant horizontal vector fields span a subbundle of the tangent bundle, called the horizontal bundle HN Generalized contact mappings f on N are smooth mappings which preserve HN The question is: how many such mappings exist? In the case of the Heisenberg group these are the contact mappings in the classical sense and they exist in abundance In this paper it is shown that if the dimension of the center of N is at least three, then the generalized contact mappings are in the automorphism group of a finite dimensional Lie algebra g The elements in ${\vec g}$ are the infinitesimal generators of local one parameter subgroups of generalized contact transformations Rigidity is defined as the property that ${\vec g}$ is finite dimensional For the case of the complexified Heisenberg group, ie the case when the dimension of the center of N is two, it has been shown [RR] that g is infinite dimensional

H. M. Reimann

Papers

Foundations for the Theory of Quasiconformal Mappings on the Heisenberg Group

Quasiconformal mappings on the Heisenberg group

Functions of bounded mean oscillation and quasiconformal mappings

Ordinary Differential Equations and Quasiconformal Mappings.

Rigidity of H-type groups