In this chapter we introduce and study Sobolev spaces of functions defined on arbitrary metric measure spaces with values in a Banach space. Central to this discussion is the theory of upper gradients developed in the previous chapter. We also discuss capacity. Capacity is an outer measure on a given metric measure space, defined with the aid of the Sobolev norm, and is used in this book to describe the pointwise behavior of functions in Sobolev and Dirichlet classes.

We assume throughout this chapter that X = (X, d, μ) is a metric measure space as defined in Section 3.3, and that V is a Banach space. We also assume that 1 ≤ p < ∞ unless otherwise specifically stated.

Vector-valued Sobolev functions on metric spaces

The theory of weak upper gradients developed in Sections 6.2 and 6.3 replaces the theory of weak or distributional derivatives in the construction of Sobolev classes of functions on metric measure spaces. Note that in Sections 6.2 and 6.3 we mostly studied maps with values in an arbitrary metric space. To obtain a linear function space, we need to have a linear target; moreover, for reasons of measurability and Bochner integration, we need to assume that we have a Banach space.

Dirichlet classes

In Section 6.1 we introduced the Dirichlet space L1, p (Ω) as the space of those locally integrable functions on an open set Ω in ℝn that have distributional derivatives in Lp(Ω). The importance of Dirichlet spaces lies in the fact that imposing the p-integrability condition for a function, in addition to its gradient, is sometimes an unnecessarily strong requirement. We next discuss analogs of Dirichlet spaces of functions defined on metric measure spaces.

The Dirichlet space, or Dirichlet class, D1, p (X : V) consists of all measurable functions u : X → V that possess a p-integrable p-weak upper gradient in X. For brevity, we set D1,p(X) := D1,p(X : ℝ).

A measurable function u : X → V belongs to D1,p(X : V) if and only if it possesses a p-integrable upper gradient (Lemma 6.2.2).