A note on the Lp-Sobolev inequality

The usual Sobolev inequality in $\mathbb{R}^N$, asserts that $\|\nabla u\|_{L^p(\mathbb{R}^N)} \geq \mathcal{S}\|u\|_{L^{p^*}(\mathbb{R}^N)}$ for $1<p<N$ and $p^*=\frac{pN}{N-p}$, 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 $\Omega\subset \mathbb{R}^N$ with finite measure is established, i.e., for $\frac{2N}{N+1}<p<N$ there exists a constant $\mathcal{C}>0$ independent of $\Omega$ such that $$\|\nabla u\|^p_{L^p(\Omega)} -\mathcal{S}^p\|u\|^p_{L^{p^*}(\Omega)} \geq \mathcal{C}|\Omega|^{-\frac{\gamma}{p^*(p-1)}} \|u\|_{L^{\bar{p}}_w(\Omega)}^{\gamma}\| u\|_{L^{p^*}(\Omega)}^{p-\gamma},\quad \mbox{for all}\ u\in C^\infty_0(\Omega)\setminus\{0\},$$ where $\gamma=\max\{2,p\}$, $\bar{p}=p^*(p-1)/p$, and $\|\cdot\|_{L^{\bar{p}}_w(\Omega)}$ denotes the weak $L^{\bar{p}}$-norm. Moreover, we establish a sharp upper bound of Sobolev inequality in $\mathbb{R}^N$.