Abstract
With a view towards Riemannian or sub-Riemannian manifolds, RCD metric spaces and specially fractals, this paper proves Sobolev embedding theorems in the general framework of Dirichlet spaces. Under suitable assumptions that are verified in a variety of settings, we obtain the whole family of Gagliardo-Nirenberg and Trudinger-Moser inequalities with optimal exponents. These turn out to depend not only on the Hausdorff and walk dimensions of the space but also on other invariants. In addition, we prove Morrey type inequalities and apply them to study the infimum of the exponents that ensure continuity of Sobolev functions. The results are illustrated in the case of fractals with the Vicsek set, whereas several conjectures are made for general nested fractals and the Sierpinski carpet.
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