On a class of interpolation inequalities on the 2D sphere

TL;DR

This paper establishes sharp interpolation inequalities on the 2D sphere, providing precise constants for Sobolev embeddings and analyzing orthonormal systems of functions and divergence-free vector fields.

Contribution

It introduces new estimates for $L^p$-norms of orthonormal systems on the 2D sphere and derives sharp constants in Gagliardo--Nirenberg inequalities.

Findings

Sharp constants in Sobolev embeddings on the 2D sphere

New $L^p$ estimates for orthonormal systems

Analysis of divergence-free vector functions

Abstract

We prove estimates for the $L^p$-norms of systems of functions and divergence free vector functions that are orthonormal in the Sobolev space $H^1$ on the 2D sphere. As a corollary, order sharp constants in the embedding $H^1\hookrightarrow L^q$, $q<\infty$, are obtained in the Gagliardo--Nirenberg interpolation inequalities.