Green's function asymptotics and sharp interpolation inequalities

S. V. Zelik ; A. A. Ilyin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 69 (2014) no. 2, pp. 209-260

Résumé

A general method is proposed for finding sharp constants for the embeddings of the Sobolev spaces $H^m(\mathscr{M})$ on an $n$-dimensional Riemannian manifold $\mathscr{M}$ 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 $\mathbb{T}^n$ and the $n$-dimensional sphere $\mathbb{S}^n$ are treated in detail, as well as certain manifolds with boundary. In certain cases when $\mathscr{M}$ 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. Bibliography: 28 titles.

Détail

Keywords: Sobolev inequalities, interpolation inequalities, Green's function, Carlson inequality.
Mots-clés : sharp constants