Green's function asymptotics and sharp point-wise interpolation inequalities

Abstract: We propose a general method for finding sharp constants in the imbeddings of the Hilbert Sobolev spaces of order m defined on a n-dimensional Riemann manifold into the space of bounded continuous functions, where m>n/2. The method is based on the analysis of the asymptotics with respect to the spectral parameter of the Green's function of the elliptic operator of order 2m, the domain of the square root of which defines the norm of the corresponding Sobolev space. The cases of the n-dimensional torus and n-dimensional sphere are treated in detail, as well as some manifolds with boundary. In certain cases when the manifold is compact, multiplicative inequalities with remainder terms of various types are obtained. Inequalities with correction term for periodic functions imply an improvement for the well-known Carlson inequalities.