关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

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

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An Introduction To The Theory Of Point Processes

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

Weighted Inequalities for Fractional Integrals on Euclidean and Homogeneous Spaces

In this paper, we fully extend to the Heisenberg group endowed with its intrinsic Carnot-Caratheodory metric and perimeter the classical De Giorgi's rectifiability divergence theorems.

https://link.springer.com/content/pdf/10.1007/s002080100228.pdf

Rectifiability and perimeter in the Heisenberg group

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/pdf/holder-regularity-theorem-for-a-class-of-linear-nonuniformly-115vdt0qzb.pdf

Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients

In this article we study codimension 1 rectifiable sets in Carnot groups and we extend classical De Giorgi ’s rectifiability and divergence theorems to the setting of step 2 groups Related problems in higher step Carnot groups are discussed, pointing on new phenomena related to the blow up procedure

On the structure of finite perimeter sets in step 2 Carnot groups

Existence and Uniqueness of Nonnegative Solutions of Quasilinear Equations inRn

Dans cet article nous comparons les differentes definitions qui ont ete donnees de l'espace de Sobolev associe a un espace metrique qui n'admet aucune structure differentielle. Nous prouvons en particulier que l'espace de Sobolev W 1,p qu'on obtient a partir de la metrique de Carnot-Caratheodory associee a une famille de champs de vecteurs {X 1 ,...,X m } coincide pour p > 1 avec l'espace naturel des fonctions u ∈ L p telles que X j u ∈ L p pour j = 1,..., m lorsque toute fonction lipschitzienne satisfait une inegalite de Poincare intrinseque, convenable.

/pdf/definitions-of-sobolev-classes-on-metric-spaces-2cs195yq2v.pdf

Bruno Franchi

Papers

Rectifiability and perimeter in the Heisenberg group

Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients

On the structure of finite perimeter sets in step 2 Carnot groups

Existence and Uniqueness of Nonnegative Solutions of Quasilinear Equations inRn

Definitions of sobolev classes on metric spaces