On BV functions and essentially bounded divergence-measure fields in metric spaces

Sobolev and Bounded Variation Functions on Metric Measure Spaces

Contents Chapter 1. Introduction 1 1. History 1 2. Motivations 3 3. Examples of metric measure spaces 4 Chapter 2. H-Sobolev space and first tools of differential calculus 9 1. Relaxed slope and Cheeger energy 9 2. Elements of differential calculus 11 3. Reminders of convex analysis 14 4. Laplacian and integration by parts formula 15 5. Heat flow in (X, d, m) 16 Chapter 3. The Lagrangian (Beppo...

Sobolev and BV spaces on metric measure spaces via derivations and integration by parts

We develop a theory of BV and Sobolev Spaces via integration by parts formula in abstract metric spaces; the role of vector fields is played by Weaver’s metric derivations. The definition hereby given is shown to be equivalent to many others present in literature. Introduction In the last few years a great attention has been devoted to the theory of Sobolev spaces W 1,q on metric measure spaces...

Functions on Localized BMO Spaces over Doubling Metric Measure Spaces

Let X be a doubling metric measure space. In this paper, the authors introduce the notions of Property (M̃) and Property (P ) of X , prove that Property (M̃) implies Property (P ) and give some equivalent characterizations of Property (M̃) and Property (P ). If X has Property (P ), the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the se...

DIFFERENTIABILITY OF p-HARMONIC FUNCTIONS ON METRIC MEASURE SPACES

We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz co...

Equivalent definitions of BV space and of total variation on metric measure spaces