A Note About a Caffarelli-Kohn-Nirenberg Type Inequality for Euclidean Submanifolds

摘要

In the seminal paper, Michael and Simon (Comm Pure Appl Math 26:361-379, 1973) generalized the celebrated Sobolev inequality for submanifolds into Euclidean spaces. The universal constant obtained by them does not depend on the submanifold. A sort of applications to the submanifold theory can be derived from that inequality. Batista et al. (J Differ Equ 263:5813-5829, 2017) the authors proved an optimal Hardy-Sobolev inequality for Euclidean submanifolds, in the same vein as Carron (J Math Pures Appl 76(10):883-891, 1997). In this paper, we prove a weighted Michael-Simon-Sobolev and a Caffarelli-Kohn-Nirenberg inequality, as well as some of their derivatives, such as Gagliardo-Nirenberg and Nash inequalities, for submanifolds in Euclidean spaces.