Sobolev function

Definition

Let $M$ be a differential manifold which is also a measured manifold, viz it is equipped with a measure. A function $f$ from $M$ to $\mathbb {R}$ is said to be a Sobolev function of type $(k,p)$ if the function, and all its first $k$ derivatives, are in $L^{p}$, and the total integral of these functions over $M$ is finite.

The set of all Sobolev functions on a differential manifold forms a vector space, and this is denoted as the Sobolev space $H^{k,p}$.