Extremal values of the (fractional) Weinstein functional on the hyperbolic space

TL;DR

This paper investigates extremal functions for Weinstein functionals on hyperbolic space, showing their maximum values match those on Euclidean space and proving the non-existence of maximizers in certain Sobolev spaces.

Contribution

It proves that the maximal Weinstein functional value on hyperbolic space equals that on Euclidean space and confirms the non-attainment of maximizers, extending results to fractional Laplacians.

Findings

Maximum Weinstein functional value on hyperbolic space equals that on Euclidean space.

No extremal functions are attained in the Sobolev space $H^1( ext{hyperbolic space})$.

Results extend to fractional Laplacian Weinstein functionals.

Abstract

We make a study of Weinstein functionals, first defined in ~\cite{W}, on the hyperbolic space $\mathbb{H}^n$. We are primarily interested in the existence of Weinstein functional maximisers, or, in other words, existence of extremal functions for the best constant of the Gagliardo-Nirenberg inequality. The main result is that the maximum value of the Weinstein functional on $\mathbb{H}^n$ is the same as that on $\mathbb{R}^n$ and the related fact that the maximum value of the Weinstein functional is not attained on $\mathbb{H}^n$, when maximisation is done in the Sobolev space $H^1(\mathbb{H}^n)$. This proves a conjecture made in ~\cite{CMMT} and also answers questions raised in several other papers (see, for example, ~\cite{B}). We also prove that a corresponding version of the conjecture will hold for the Weinstein functional with the fractional Laplacian as well.