The Gagliardo-Nirenberg-Sobolev inequality for $1<p<n$

The smooth real case. Let $u\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n;\mathbb{R})$ and choose $\chi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ with $0\le \chi \le 1$ and $\chi =1$ on a neighborhood of $\mathrm{supp}u$. Put $s:=\mathrm{\mid }u\mathrm{\mid }$, $q:=n\mathrm{/}(n-1)$, and $v_{\delta }:=\chi (u^2+{\delta }^2{)}^{\gamma \mathrm{/}2}$ for $0<\delta \le 1$. Then $v_{\delta }\in C_c^{\mathrm{\infty }}$. By [F6], on $\mathrm{supp}u$ its gradient has modulus $M_{\delta }:=\gamma s(s^2+{\delta }^2{)}^{(\gamma -2)\mathrm{/}2}\mathrm{\mid }Du\mathrm{\mid }$, while off that support the only derivative term is ${\delta }^{\gamma }D\chi$. Thus [F2] gives $\mathrm{\parallel }{v}_{\delta }{\mathrm{\parallel }}_{L^q}\le C_1(n)(\int M_{\delta }+{\delta }^{\gamma }\int \mathrm{\mid }D\chi \mathrm{\mid })$.

Let ${\delta }_j\downarrow 0$. Since $v_{{\delta }_j}\to \mathrm{\mid }u{\mathrm{\mid }}^{\gamma }$ pointwise and is dominated by the bounded compactly supported function $\chi (s^2+1{)}^{\gamma \mathrm{/}2}$, [F7] gives $\mathrm{\parallel }{v}_{{\delta }_j}{\mathrm{\parallel }}_{L^q}\to \mathrm{\parallel }u{\mathrm{\parallel }}_{L^{p^{\ast }}}^{\gamma }$. Also $M_{{\delta }_j}\to \gamma s^{\gamma -1}\mathrm{\mid }Du\mathrm{\mid }$ almost everywhere. If $1<\gamma \le 2$, then $M_{\delta }\le \gamma s^{\gamma -1}\mathrm{\mid }Du\mathrm{\mid }$; if $\gamma >2$, then for $0<\delta \le 1$, $M_{\delta }\le C_{\gamma }(s^{\gamma -1}+s)\mathrm{\mid }Du\mathrm{\mid }$. These majorants are integrable because $u$ is smooth and compactly supported, so [F7] gives $\int M_{{\delta }_j}\to \gamma \int s^{\gamma -1}\mathrm{\mid }Du\mathrm{\mid }$, while ${\delta }_j^{\gamma }\int \mathrm{\mid }D\chi \mathrm{\mid }\to 0$. Taking limits in the endpoint estimate yields $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{p^{\ast }}}^{\gamma }\le C_1(n)\gamma \int s^{\gamma -1}\mathrm{\mid }Du\mathrm{\mid }$.

Holder [F3] and $(\gamma -1)p\mathrm{/}(p-1)=p^{\ast }$ [F1] give $\int s^{\gamma -1}\mathrm{\mid }Du\mathrm{\mid }\le \mathrm{\parallel }u{\mathrm{\parallel }}_{L^{p^{\ast }}}^{\gamma -1}\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^p}$. If $u\mathrm{\ne }0$ in $L^{p^{\ast }}$, divide by $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{p^{\ast }}}^{\gamma -1}$; if $u=0$ almost everywhere, the estimate is immediate. Thus the real smooth case holds with constant $C_1(n)\gamma$.

The smooth complex case. Let $u\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n;\mathbb{C})$ with real and imaginary parts $a$ and $b$. Applying step 2.1 to both parts and using $\mathrm{\mid }u\mathrm{\mid }\le \mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }$ gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{p^{\ast }}}\le C_1(n)\gamma (\mathrm{\parallel }Da{\mathrm{\parallel }}_{L^p}+\mathrm{\parallel }Db{\mathrm{\parallel }}_{L^p})\le 2C_1(n)\gamma \mathrm{\parallel }Du{\mathrm{\parallel }}_{L^p}$.

Passage to $W^{1,p}$ by density. Let $u\in W^{1,p}({\mathbb{R}}^n;\mathbb{K})$. By [F4] choose $u_k\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n;\mathbb{K})$ with $u_k\to u$ in $W^{1,p}({\mathbb{R}}^n)$. Applying step 2.1 (real case) or step 3.1 (complex case) to the differences gives $\mathrm{\parallel }{u}_k-u_j{\mathrm{\parallel }}_{L^{p^{\ast }}}\le C^{\mathrm{\prime }}(n,p)\mathrm{\parallel }D(u_k-u_j){\mathrm{\parallel }}_{L^p}$, so $(u_k)$ is Cauchy in $L^{p^{\ast }}$. By [F5] it converges in $L^{p^{\ast }}$ to a class $w$, and a subsequence converges to $w$ almost everywhere. Since $u_k\to u$ in $L^p$, a further subsequence converges to $u$ almost everywhere, so $w=u$ almost everywhere. Passing to the limit in the smooth inequality gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{p^{\ast }}}\le C^{\mathrm{\prime }}(n,p)\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^p}$, and renaming this constant proves the claim.