A note on the Lp-Sobolev inequality

Abstract: 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 $\mathbb{R}^N$.