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

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 $$0.0844 \leq \mathcal{K}_{4}(\mathbb{S}^{4})\leq 0.1728,$$ and the constant $\mathcal{K}_{4}(\mathbb{T}^{4})$ of the Lieb-Thirring inequality on the torus $\mathbb{T}^{4}$ satisfies $$0.0190 \leq \mathcal{K}_{4}(\mathbb{T}^{4}) \leq 0.1222.$$