Newtonian space

Summary

The Newtonian space $N^{1,p}(X)$ is a version of the Sobolev space $W^{1,p}$ for functions on a metric measure space $X$. It replaces the gradient by an upper gradient, a function that bounds the change of $u$ along every rectifiable curve. It was introduced by Nageswari Shanmugalingam, and it is a Banach space for $1 \le p < \infty$. It is the basic function space of nonlinear potential theory on metric spaces, where it carries $p$-capacities and p-harmonic functions.