Gagliardo-Nirenberg, Trudinger-Moser and Morrey inequalities on Dirichlet spaces

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 variety settings, we obtain whole family Gagliardo-Nirenberg Trudinger-Moser inequalities with optimal exponents. These turn out to depend not only on Hausdorff walk dimensions space but also other invariants. In addition, prove Morrey type apply them study infimum exponents ensure continuity functions. The results illustrated case fractals Vicsek set, whereas several conjectures made for nested Sierpinski carpet.