Sharp k-order Sobolev inequalities in the hyperbolic space ${\Bbb H}^n$

TL;DR

This paper establishes the sharp k-th order Sobolev inequalities in hyperbolic space for all positive integers k, resolving an open question and confirming the optimality of the associated constants.

Contribution

It provides the first complete set of sharp Sobolev inequalities in hyperbolic space for all orders, answering a longstanding open problem.

Findings

Derived sharp k-th order Sobolev inequalities in hyperbolic space.

Proved the optimality of the Sobolev constants.

Extended results to all k=1,2,3,... in hyperbolic geometry.

Abstract

In this paper, we obtain the sharp $k$-th order Sobolev inequalities in the hyperbolic space ${\H}^n$ for all $k=1,2,3,\cdots$. This gives an answer to an open question raised by Aubin in [5, p.$\;$176-177] for $W^{k,2}({\H}^n)$ with $k>1$. In addition, we prove that the associated Sobolev constants are optimal.