The Gagliardo-Nirenberg-Sobolev inequality in $\mathbb{R}^n$ is a classical Sobolev embedding with many applications in the theory of PDEs and calculus of variations. This inequality asserts that, for any $1\leq p
In this talk we present the full scale of Gagliardo-Nirenberg-Sobolev inequalities in the more general framework of Dirichlet spaces with (sub-)Gaussian heat kernel estimates. In particular, we will discover that the optimal exponent not only depends on the Hausdorff dimension of the underlying space, but also on other invariants. To this end, we will discuss a recent approach to $(1,p)$-Sobolev spaces via heat semigroups inspired by ideas that go back to work of de Giorgi and Ledoux.
Besides heat kernel estimates, the main assumption on the underlying space is a non-negative curvature type condition that we call weak Bakry-\'Emery. This condition is satisfied in classical settings as well as in fractals like (infinite) Sierpinski gaskets and carpets. If time permits, we will outline some results and conjectures concerning further Sobolev embeddings and introduce a certain critical exponent that might be related to other dimensions of interest in the theory of metric measure spaces.
This talk is based on joint work with F. Baudoin.