With a view towards Riemannian or sub-Riemannian manifolds, RCD metric spaces and specially fractals, this paper makes a step further in the development of a theory of heat semigroup based $(1,p)$ Sobolev spaces in the general framework of Dirichlet spaces. Under suitable assumptions that are verified in a variety of settings, the tools developed by D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste in the paper "Sobolev inequalities in disguise" allow us to obtain the whole family of Gagliardo-Nirenberg and Trudinger-Moser inequalities with optimal exponents. The latter 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 for fractals using the Vicsek set, whereas several conjectures are made for nested fractals and the Sierpinski carpet.

中文翻译：

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Dirichlet 空间上的 Gagliardo-Nirenberg、Trdinger-Moser 和 Morrey 不等式

本文着眼于黎曼或亚黎曼流形、RCD 度量空间和特殊分形，在 Dirichlet 空间的一般框架中进一步发展基于热半群的 $(1,p)$ Sobolev 空间理论. 在各种设置中得到验证的适当假设下，由 D. Bakry、T. Coulhon、M. Ledoux 和 L. Saloff-Coste 在“Sobolev inequalities in disguise”论文中开发的工具使我们能够获得具有最优指数的 Gagliardo-Nirenberg 和 Trudinger-Moser 不等式。后者不仅取决于空间的 Hausdorff 和步行维度，还取决于其他不变量。此外，我们证明了 Morrey 型不等式并将其应用于研究确保 Sobolev 函数连续性的指数的下界。