Reverse conformally invariant Sobolev inequalities on the sphere

TL;DR

This paper investigates the sharp constants in conformally invariant Sobolev inequalities on the sphere for noninteger orders, extending previous results and confirming their validity across all dimensions n ≥ 2.

Contribution

It extends existing Sobolev inequality results to noninteger orders and confirms their validity in all dimensions n ≥ 2.

Findings

Extended Sobolev inequalities to noninteger order s.

Confirmed validity of inequalities in all dimensions n ≥ 2.

Provided sharp constants for the inequalities.

Abstract

We consider the optimization problem corresponding to the sharp constant in a conformally invariant Sobolev inequality on the $n$-sphere involving an operator of order $2s> n$. In this case the Sobolev exponent is negative. Our results extend existing ones to noninteger values of $s$ and settle the question of validity of a corresponding inequality in all dimensions $n\geq 2$.