---- > [!theorem] Theorem. ([[Gagliardo-Nirenberg-Sobolev Inequality]]) > Assume $1 \leq p <n$. Let $p^{*}=\frac{np}{n-p}$ be the [[Sobolev conjugate]] of $p$. > > > Then there exists a constant $C_{n,p}$ such that $\|u\|_{p^{*}} \leq C_{n,p} \| D u\|_{p}$ > [[Sobolev space|for all]] $u \in W^{1, p}(\mathbb{R}^{n})$. > > It follows that there is a [[continuous]] inclusion of sets [^1] $W^{1,p}(\mathbb{R}^{n}) \hookrightarrow L^{p^{*}}(\mathbb{R}^{n}).$ > Moreover, a scaling argument shows that if $q$ is such that $\|u\|_{p^{*}} \leq C_{n,p}\|D u\|_{p} \text{ for all } u \in W^{1,p}(\mathbb{R}^{n})$ for some $1 \leq p < n$, then necessarily $q=p^{*}$. [^1]: If $u \in W^{1,p}$, meaning $\|u\|_{1,p}^{p}= \|u\|_{p}^{p}+ \|Du\|_{p}^{p}<\infty$, then so too $\|Du\|_{p}^{p}<\infty$, hence $\|u\|_{p^{*}} \leq C_{n,p}\|Du\|_{p}<\infty$. Thus $u \in L^{p^{*}}$. Moreover, the inclusion is [[continuous]] because $\|u\|_{p^{*}} \leq C_{n,p} \|Du\|_{p} \leq C_{n,p} \|u\|_{1,p}$. > [!proof]- Proof. ([[Gagliardo-Nirenberg-Sobolev Inequality]]) > **Claim 1.** Enough to prove for $u \in C_{c}^{\infty}(\mathbb{R}^{n})$. Indeed, suppose GNS holds for [[continuously differentiable|smooth]] [[compact|compactly]] [[support|supported]] elements of $W^{1,p}$. Let $u \in W^{1,p}(\mathbb{R}^{n})$. Since $p < \infty$, [[Sobolev density of smooth compactly supported functions|we know]] $C^{\infty}_{c}(\mathbb{R}^{n})$ is [[dense]] in $W^{1,p}(\mathbb{R}^{n})$. Thus there exists a [[sequence]] $(u_{m}) \subset C_{c}^{\infty}(\mathbb{R}^{n})$ such that $u_{m} \to u$ in $W^{1,p}$. Meaning $\|u_{m} - u\|_{p}^{p} + \|Du_{m}- Du\|_{p}^{p} \to 0$. Note this means $\|Du_{m}\|_{p} \to \|Du\|_{p}$. > > By assumption, $\|u_{m}\|_{p^{*}} \leq C_{n,p,m}\|Du_{m}\|_{p}$. Can we pass to limits? Yes on RHS, as we just saw $\|Du_{m}\|_{p} \to \|Du\|_{p}$. LHS have to be careful! Have $p^{*}$, *not* $p$. Indeed, it may not hold that $\|u_{m}\|_{p^{*}} \to \|u\|_{p^{*}}$ But we don't actually need this: it's enough [[limit inferior and limit superior|that]] $\|u\|_{p^{*}} \leq \liminf_{m \to \infty} \|u_{m}\|_{p^{*}}$. [[Fatou's Lemma]] gives the claim. > > Have $u_{m} \to u$ in $W^{1,p}$, thus $u_{m} \to u$ [[Lp-norm|in]] $L^{p}$, [[convergence in measure|thus]] have $u_{m} \to u$ [[almost-everywhere]] after dropping to a [[subsequence]]. TODO finish, not sure why i stopped. > > > **Claim 2.** We prove only for case $p=1$, $n=2$ because the general case doesn't require any bigger ideas than this one. $p^{*}=2$, thus the goal is to show $\|u\|_{2} \leq C \|Du\|_{1} \text{ for all } u \in C^{\infty}_{c}(\mathbb{R}^{2}).$ > Define maps $f_{x_{1}}:\mathbb{R} \to \mathbb{R}$, $f_{x_{1}}(x_{2})=u(x_{1},x_{2})$ and similarly $f_{x_{2}}(x_{1})=u(x_{1},x_{2})$. By [[The Fundamental Theorem of Calculus]] (applicable because of [[compact]] [[support]]), $u(x_{1},x_{2})=f_{x_{2}}(x_{1})=\int _{-\infty}^{x_{1}} \partial_{1}u(y_{1},x_{2}) \, dy_{1}.$ > Thus $\begin{align} > |u(x_{1},x_{2})|& \leq \int _{-\infty}^{x_{1}} | \partial_{1}u(y_{1},x_{2})| \, dy_{1} \\ > & \leq \int _{-\infty} ^{\infty} |Du(y_{1}, x_{2})| \, dy_{1}. > \end{align}$Symmetrically, $|u(x_{1},x_{2})| \leq \int _{-\infty}^{\infty} |Du(x_{1},y_{2})| \, dy_{2}.$ > Now $\begin{align} > |u(x_{1},x_{2})|^{2} & \leq ( \int _{-\infty} ^{\infty} |Du(y_{1}, x_{2})| \, dy_{1}) ( \int _{-\infty}^{\infty} |Du(x_{1},y_{2})| \, dy_{2}) , > \end{align}$ > thus $\begin{align} > \|u\|_{2}^{2} &= \int _{\mathbb{R}^{2}} |u(x_{1},x_{2})|^{2}\, \\ > &\leq \int _{-\infty}^{\infty} \int _{-\infty}^{\infty} |u(x_{1},x_{2})|^{2} \, dx_{1} \, dx_{2} \\ > &\leq \int _{-\infty}^{\infty} \int _{-\infty}^{\infty} \left( \int _{-\infty} ^{\infty} |Du(y_{1}, x_{2})| \, dy_{1} \right) \left( \int _{-\infty}^{\infty} |Du(x_{1},y_{2})| \, dy_{2} \right) \,dx_{1} \, dx_{2} \\ > &= \underbrace{ \left( \int _{-\infty}^{\infty} \int _{-\infty} ^{\infty} |Du(y_{1}, x_{2})| \, dy_{1} \, dx_{2} \right) }_{ \|Du\|_{1} }\underbrace{ \left( \int _{-\infty}^{\infty} \int _{-\infty}^{\infty} |Du(x_{1},y_{2})| \, dy_{2} \,dx_{1} \right) }_{ \|Du\|_{1} } \\ > &= \|Du\|_{1}^{2}, > \end{align}$ > where the last step used [[Fubini's Theorem]]. The inequality follows. > > ^3c6d4a ---- #### > [!proof] Proof that $q=p^{*}$. > **Proof that $q=p^{*}$.** Suppose $q$ is such that $\|u\|_{q} \leq C_{n,p} \|\nabla u\|_{p}$ for some $C_{n,p}>0$, for all $u \in W^{1,p}(\mathbb{R}^{n})$. For $u \in C^{\infty}_{c}(\mathbb{R}^n)$ and $\lambda>0$ define $u_{\lambda}(x):=u(\lambda x)$; note $u_{\lambda} \in C_{c}^{\infty}(\mathbb{R}^{n})$. We have $\|u_{\lambda}\|_{q} \leq C_{p,q,n} \|\nabla u _{\lambda}\|_{p}\text{ for all } u \in C^{\infty}_{c}(\mathbb{R}^{n}).$ > Let's compute the LHS and RHS. Have $\begin{align} > \|u_{\lambda}\|_{q}^{q} &= \int _{\mathbb{R}^{n}} |u(\lambda x)|^{q} \, dx = \frac{1}{\lambda^{n}} \int_{\mathbb{R}^{n}} |u(x)|^{q} \, dx = \lambda^{-n} \|u\|_{q}^{q} > \end{align}$ > and $\|\nabla u_{\lambda}\|^{p}_{p}=\int _{\mathbb{R}^{n}} | \lambda (\nabla u) (\lambda x) |^{p} \, dx = \frac{\lambda^{p}}{\lambda^{n}} \int _{\mathbb{R}^{n} } |\nabla u(x)|^{p} \, dx = \lambda^{p-n} \|\nabla u\|_{p}^{p} $ > where we have used [[change of variables under a linear transformation]] with $h=|\_|^{\cdot}$. Plugging in, we obtain $\lambda^{-n/q}\|u\|^{}_{q} \leq C_{n,p} \lambda^{(p-n)/p} \|\nabla u\|_{p}^{},$ > that is, $\|u\|_{q} \leq C_{n,p}\lambda^{1-n/p + n/q} \|\nabla u\|_{p}.$ > Now, if $1-\frac{n}{p}+\frac{n}{q} \neq 0$ then the RHS can be made arbitrarily small or small by sending $\lambda$ either to $0$ or $\infty$, which is impossible when $u \not \equiv 0$. Thus $1-\frac{n}{p}+\frac{n}{q}=0$. ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```