A new approach to Sobolev spaces in metric measure spaces

作者：Sjodin Tomas^*

来源：Nonlinear Analysis-Theory Methods & Applications, 2016, 142: 194-237.

DOI：10.1016/j.na.2016.04.010

• Sobolev space
• Metric measure space
• Mass transport

Let (X, d(X), mu) be a metric measure space where X is locally compact and separable and mu is a Borel regular measure such that 0 < mu(B(x, r)) < infinity for every ball B(x, r) with center x is an element of X and radius r > 0. We define chi to be the set of all positive, finite non- zero regular Borel measures with compact support in X which are dominated by mu, and M = X boolean OR {0}. By introducing a kind of mass transport metric d(M) on this set we provide a new approach to first order Sobolev spaces on metric measure spaces, first by introducing such for functions F : X -> R, and then for functions f : X -> [-infinity, infinity] by identifying them with the unique element F-f : X -> R defined by the mean- value integral:
Ff(eta) - 1/vertical bar vertical bar eta vertical bar vertical bar integral f d eta.
In the final section we prove that the approach gives us the classical Sobolev spaces when we are working in open subsets of Euclidean space R-n with Lebesgue measure.

• 出版日期2016-9