Sobolev type inequalities for doubling metric measure spaces

Abstract

Consider a doubling metric measure space with a notion of gradient. One writes two scales of adapted global Nash and Gagliardo-Nirenberg $L^p$ inequalities with different geometric contents as $p$ varies and one studies their relationship. Under some suitable assumptions on the associated operator, the $L^2$ version is equivalent to a heat kernel upper bound. This relies on joint works with Salahaddine Boutayeb and Adam Sikora.