Green's function asymptotics and sharp interpolation inequalities

A general method is proposed for finding sharp constants for the embeddings of the Sobolev spaces H (M) on an n-dimensional Riemannian manifold M into the space of bounded continuous functions, where m > n/2. The method is based on an analysis of the asymptotics with respect to the spectral parameter of the Green's function of an elliptic operator of order 2m whose square root has domain determining the norm of the corresponding Sobolev space. The cases of the n-dimensional torus T and the n-dimensional sphere S are treated in detail, as well as certain manifolds with boundary. In certain cases when M is compact, multiplicative inequalities with remainder terms of various types are obtained. Inequalities with correction terms for periodic functions imply an improvement for the well-known Carlson inequalities. © 2014 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd