论文标题
Gagliardo-Nirenberg，Trudinger-Moser和Morrey的不平等现象
Gagliardo-Nirenberg, Trudinger-Moser and Morrey inequalities on Dirichlet spaces
论文作者
论文摘要
为了观察Riemannian或亚黎曼分歧，RCD度量空间和特别是分形，本文在Dirichlet Space的一般框架中的基于热半群的$（1，P）$ SOBOLEV空间的发展方面迈出了一步。在各种环境中验证的合适假设下，D。Bakry，T。Coulhon，M。Ledoux和L. Saloff-Coste开发的工具在论文中“ Sobolev不平等”中的不平等现象“使我们能够获得整个Gagliardo-Nirenberg和Trudinger-Moser不平等的家族。后者不仅取决于豪斯多夫和空间的步行尺寸，还取决于其他不变的尺寸。此外，我们证明了Morrey型不平等，并将其应用于研究确保Sobolev函数连续性的指数的最大值。使用Vicsek组为分形说明了结果，而对嵌套分形和Sierpinski地毯做出了几种猜想。
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.
