Korevaar-Schoen-Sobolev spaces and critical exponents in metric measure spaces

Abstract

We present developments in the theory of Korevaar-Schoen-Sobolev spaces on metric measure spaces. While this theory coincides with those of Cheeger and Shanmugalingam if the space is doubling and satisfies a Poincar\'e inequality, it offers new perspectives in the context of fractals for which the approach by weak upper gradients is inadequate.