A note on the Lp-Sobolev inequality

Abstract

The usual Sobolev inequality in ${\mathbb{R}}^N$, asserts that $\Vert \nabla u{\Vert }_{L^p({\mathbb{R}}^N)}\ge \mathcal{S}\Vert u{\Vert }_{L^{p^{\ast }}({\mathbb{R}}^N)}$ for $1<p<N$ and $p^{\ast }=\frac{pN}{N-p}$, with $\mathcal{S}$ being the sharp constant. Based on a recent work of Figalli and Zhang [Duke Math. J., 2022], a weak norm remainder term of Sobolev inequality in a subdomain $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ with finite measure is established, i.e., for $\frac{2N}{N+1}<p<N$ there exists a constant $\mathcal{C}>0$ independent of $\hat{I}\text{c◯}$ such that \[ \|\nabla u\|^p_{L^p(Ω)} -\mathcal{S}^p\|u\|^p_{L^{p^*}(Ω)} \geq \mathcal{C}|Ω|^{-\fracγ{p^*(p-1)}} \|u\|_{L^{\bar{p}}_w(Ω)}^γ\| u\|_{L^{p^*}(Ω)}^{p-γ},\quad \mbox{for all}\ u\in C^\infty_0(Ω)\setminus\{0\}, \] where ${\hat{I}}^3=\max \{2,p\}$, $\bar{p}=p^{\ast }(p-1)/p$, and $\Vert \cdot {\Vert }_{L_w^{\bar{p}}(\hat{I}\text{c◯})}$ denotes the weak $L^{\bar{p}}$-norm. Moreover, we establish a sharp upper bound of Sobolev inequality in ${\mathbb{R}}^N$.