Analysis on asymmetric metric measure spaces: $q$-heat flow, $q$-Laplacian and Sobolev spaces

Abstract

We investigate geometric analysis on metric measure spaces equipped with asymmetric distance functions. Extending concepts from the symmetric case, we introduce upper gradients and corresponding $L^q$-energy functionals as well as $q$-Laplacian in the asymmetric setting. Along the lines of gradient flow theory, we then study $q$-heat flow as the $L^2$-gradient flow for $L^q$-energy. We also study the Sobolev spaces over asymmetric metric measure spaces, extending some results of Ambrosio--Gigli--Savar\'e to asymmetric distances. A wide class of asymmetric metric measure spaces is provided by irreversible Finsler manifolds, which serve to construct various model examples by pointing out genuine differences between the symmetric and asymmetric settings.