A note on the Lp-Sobolev inequality

The usual Sobolev inequality in ${\mathbb{R}}^N$, asserts that $\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^p({\mathbb{R}}^N)}\ge \mathcal{S}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{p^{\ast }}({\mathbb{R}}^N)}$ 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 $\mathrm{\Omega }\subset {\mathbb{R}}^N$ with finite measure is established, i.e., for $\frac{2N}{N+1}<p<N$ there exists a constant $\mathcal{C}>0$ independent of $\mathrm{\Omega }$ such that $$\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}^p-{\mathcal{S}}^p\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{p^{\ast }}(\mathrm{\Omega })}^p\ge \mathcal{C}\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{-\frac{\gamma }{p^{\ast }(p-1)}}\mathrm{\parallel }u{\mathrm{\parallel }}_{L_w^{\bar{p}}(\mathrm{\Omega })}^{\gamma }\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{p^{\ast }}(\mathrm{\Omega })}^{p-\gamma },forallu\in C_0^{\mathrm{\infty }}(\mathrm{\Omega })\setminus \{0\},$$ where $\gamma =\max \{2,p\}$, $\bar{p}=p^{\ast }(p-1)\mathrm{/}p$, and $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{L_w^{\bar{p}}(\mathrm{\Omega })}$ denotes the weak $L^{\bar{p}}$-norm. Moreover, we establish a sharp upper bound of Sobolev inequality in ${\mathbb{R}}^N$.