Hypercontractivity Property and General Optimal $$L^p$$ L p -Entropy Inequalities on Riemannian Manifolds

Abstract

Let (M, g) be a n-dimensional smooth compact Riemannian manifold without boundary with $$n\ge 2$$ . We prove that the optimal Riemannian p-entropy inequality $$\begin{aligned} \int _M |u|^p\log (|u|^p) \; dv_g\le \dfrac{n}{\tau }\log \left[ {\mathcal {A}}(p,\tau )\left( \int _M |\nabla _g u|^p\; dv_g\right) ^{\frac{\tau }{p}}+{\mathcal {B}}(p)\right] \end{aligned}$$ is valid for every $$u\in H^{1,p}(M)$$ with $$\Vert u\Vert _p=1$$ where $$p>1$$ and $$1\le \tau < \min \{2,p\}$$ or $$\tau = p \le 2$$ . Also, we investigated the relationship between this optimal inequality and the hypercontractivity property for the non-linear evolution equation $$\begin{aligned} u_t=\Delta _p\left( u^{\frac{1}{p-1}}\right) ,\quad x\in M,\ t>0. \end{aligned}$$ When $$\tau =p\le 2$$ we find explicit estimates of the time asymptotic behavior of their solutions with initial data in the spaces $$L^q(M)$$ , $$1\le q<\infty $$ .