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

TL;DR

This paper explores Korevaar-Schoen-Sobolev spaces in metric measure spaces, highlighting their advantages over weak upper gradient methods especially in fractal contexts, and discusses critical exponents and theoretical developments.

Contribution

It advances the theory of Korevaar-Schoen-Sobolev spaces, providing new insights for non-doubling fractal spaces where traditional methods fail.

Findings

Equivalence with Cheeger and Shanmugalingam spaces under certain conditions

New perspectives on fractal spaces without Poincaré inequality

Development of critical exponent theory 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.