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

TL;DR

This paper develops a method to determine sharp constants in Sobolev embedding inequalities on Riemann manifolds using Green's function asymptotics, with applications to tori, spheres, and manifolds with boundary.

Contribution

It introduces a general spectral method for finding sharp embedding constants and derives new inequalities with correction terms for specific manifolds.

Findings

Derived sharp constants for Sobolev embeddings on tori and spheres.

Obtained inequalities with correction terms that improve classical results.

Applied the method to manifolds with boundary, expanding the scope of sharp inequality analysis.

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.