Lieb-Thirring inequality on the four-dimensional sphere and torus

Abstract

In this paper, we mainly study the Lieb-Thirring inequality for families of orthonormal scalar functions on the four-dimensional sphere ${\mathbb{S}}^4$ and torus ${\mathbb{T}}^4$. The bounds of all the constants involved are obtained. Specifically, we prove that the costant ${\mathcal{K}}_4({\mathbb{S}}^4)$ of the Lieb-Thirring inequality on the sphere ${\mathbb{S}}^4$ satisfies $$0.0844\le {\mathcal{K}}_4({\mathbb{S}}^4)\le 0.1728,$$ and the constant ${\mathcal{K}}_4({\mathbb{T}}^4)$ of the Lieb-Thirring inequality on the torus ${\mathbb{T}}^4$ satisfies $$0.0190\le {\mathcal{K}}_4({\mathbb{T}}^4)\le \mathrm{0.1222.}$$