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I got stuck in the following lemma:

Lemma 1: Let $B$ be the unit ball in the 4 dimensional Euclidean space. Suppose that $u\in W^{2,2}(B)$, then $e^{u}\in L^{q}$ for any $q\geq1$.

Actually, I realized that the Lemma 1 follows from Trudinger's theorem. https://en.wikipedia.org/wiki/Trudinger%27s_theorem

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Below is my little thoughts about this problem in the beginning and just want to keep it for myself.

My idea of this that fails: Note $\int_{B} e^{pu}\leq \sum_{k=1}^{\infty}\frac{p^{k}}{k!}\int_{B} |u|^{k}$. By Sobolev imbedding we know that $W^{2,2}(B)\hookrightarrow L^{k}$ for any $k\geq 1$. So we know that $$\parallel u\parallel_{L^{k}(B)}\leq C_{k} \parallel u \parallel_{W^{2,2}} $$ We can then get : $\int_{B} e^{pu}\leq \sum_{k=1}^{\infty}\frac{p^{k}}{k!} (C_{k})^{k} \parallel u \parallel_{W^{2,2}}^{k}$. So that it suffices to control $C_{k}$. But based on my calculation, $C_{k} \sim k$, from which it seems this argument would fail.

After carefully thinking about what I need, I get the following.

Lemma2: Suppose $u\in W^{2,2}(B(r_{0}))$ where $B_{r_{0}}$ is a ball with radius $r_{0}$ in the 4 dimensional Euclidean space, then $e^{u}\in L^{q}(B_{\frac{r_{0}}{2}})$ for any $q\geq 1$.

Proof: by multiplying a cut off function $\eta$ which is equal to 1 on $B_{\frac{r_{0}}{2}}$, we can apply the theorem 7.15 on Gilbert&Trudinger's book, i.e. Elliptic partial differential equations of second order (which can be also found on the link provided by Pedro Lauridsen Ribeiro in the comment ) to get $$ \int_{B_{r_{0}}}e^{(\frac{\eta u}{C_{1}||D^{1}(\eta |u|)||^{4}})^{\frac{4}{3}}}\leq C_{2}|B_{r_{0}}| $$ which then help control any $||e^{u}||_{L^{q}(B_{\frac{r_{0}}{2}})}$ for any $q\geq 1$.

I got stuck in the following lemma:

Lemma: Let $B$ be the unit ball in the 4 dimensional Euclidean space. Suppose that $u\in W^{2,2}(B)$, then $e^{u}\in L^{q}$ for any $q>1$.

As we know this is exactly the critical case of Sobolev's imbedding where we fail to get $L^{\infty}$ bounds. Any suggestion and help would be appreciated.

My idea of this that fails:

Note $\int_{B} e^{pu}\leq \sum_{k=1}^{\infty}\frac{p^{k}}{k!}\int_{B} |u|^{k}$. By Sobolev imbedding we know that $W^{2,2}(B)\hookrightarrow L^{k}$ for any $k\geq 1$. So we know that

$$\parallel u\parallel_{L^{k}(B)}\leq C_{k} \parallel u \parallel_{W^{2,2}} $$

We can then get :

$\int_{B} e^{pu}\leq \sum_{k=1}^{\infty}\frac{p^{k}}{k!} (C_{k})^{k} \parallel u \parallel_{W^{2,2}}^{k}$.

So that it suffices to control $C_{k}$. But based on my calculation, $C_{k} \sim k$, from which it seems this argument would fail.

I got stuck in the following lemma:

Lemma1: Let $B$ be the unit ball in the 4 dimensional Euclidean space. Suppose that $u\in W^{2,2}(B)$, then $e^{u}\in L^{q}$ for any $q\geq1$.

Actuall, I realized that the lemma1 follows from Trudger's theorem. https://en.wikipedia.org/wiki/Trudinger%27s_theorem

---------------------------------------------------------

Below is my little thoughts about this problem in the beginning and just want to keep it for myself.

My idea of this that fails: Note $\int_{B} e^{pu}\leq \sum_{k=1}^{\infty}\frac{p^{k}}{k!}\int_{B} |u|^{k}$. By Sobolev imbedding we know that $W^{2,2}(B)\hookrightarrow L^{k}$ for any $k\geq 1$. So we know that $$\parallel u\parallel_{L^{k}(B)}\leq C_{k} \parallel u \parallel_{W^{2,2}} $$ We can then get : $\int_{B} e^{pu}\leq \sum_{k=1}^{\infty}\frac{p^{k}}{k!} (C_{k})^{k} \parallel u \parallel_{W^{2,2}}^{k}$. So that it suffices to control $C_{k}$. But based on my calculation, $C_{k} \sim k$, from which it seems this argument would fail.

After carefully thinking about what I need, I get the following.

Lemma2: Suppose $u\in W^{2,2}(B(r_{0}))$ where $B_{r_{0}}$ is a ball with radius $r_{0}$ in the 4 dimensional Euclidean space, then $e^{u}\in L^{q}(B_{\frac{r_{0}}{2}})$ for any $q\geq 1$.

Proof: by multiplying a cut off function $\eta$ which is equal to 1 on $B_{\frac{r_{0}}{2}}$, we can apply the theorem 7.15 on Gilbert&Trudinger's book, i.e. Elliptic partial differential equations of second order (which can be also found on the link provided by Pedro Lauridsen Ribeiro in the comment ) to get $$ \int_{B_{r_{0}}}e^{(\frac{\eta u}{C_{1}||D^{1}(\eta |u|)||^{4}})^{\frac{4}{3}}}\leq C_{2}|B_{r_{0}}| $$ which then help control any $||e^{u}||_{L^{q}(B_{\frac{r_{0}}{2}})}$ for any $q\geq 1$.

I got stuck in the following lemma:

Lemma 1: Let $B$ be the unit ball in the 4 dimensional Euclidean space. Suppose that $u\in W^{2,2}(B)$, then $e^{u}\in L^{q}$ for any $q\geq1$.

Actually, I realized that the Lemma 1 follows from Trudinger's theorem. https://en.wikipedia.org/wiki/Trudinger%27s_theorem

---------------------------------------------------------

Below is my little thoughts about this problem in the beginning and just want to keep it for myself.

My idea of this that fails: Note $\int_{B} e^{pu}\leq \sum_{k=1}^{\infty}\frac{p^{k}}{k!}\int_{B} |u|^{k}$. By Sobolev imbedding we know that $W^{2,2}(B)\hookrightarrow L^{k}$ for any $k\geq 1$. So we know that $$\parallel u\parallel_{L^{k}(B)}\leq C_{k} \parallel u \parallel_{W^{2,2}} $$ We can then get : $\int_{B} e^{pu}\leq \sum_{k=1}^{\infty}\frac{p^{k}}{k!} (C_{k})^{k} \parallel u \parallel_{W^{2,2}}^{k}$. So that it suffices to control $C_{k}$. But based on my calculation, $C_{k} \sim k$, from which it seems this argument would fail.

After carefully thinking about what I need, I get the following.

Lemma2: Suppose $u\in W^{2,2}(B(r_{0}))$ where $B_{r_{0}}$ is a ball with radius $r_{0}$ in the 4 dimensional Euclidean space, then $e^{u}\in L^{q}(B_{\frac{r_{0}}{2}})$ for any $q\geq 1$.

Proof: by multiplying a cut off function $\eta$ which is equal to 1 on $B_{\frac{r_{0}}{2}}$, we can apply the theorem 7.15 on Gilbert&Trudinger's book, i.e. Elliptic partial differential equations of second order (which can be also found on the link provided by Pedro Lauridsen Ribeiro in the comment ) to get $$ \int_{B_{r_{0}}}e^{(\frac{\eta u}{C_{1}||D^{1}(\eta |u|)||^{4}})^{\frac{4}{3}}}\leq C_{2}|B_{r_{0}}| $$ which then help control any $||e^{u}||_{L^{q}(B_{\frac{r_{0}}{2}})}$ for any $q\geq 1$.

I got stuck in the following lemma:

New Lemma: Let $B$ be the unit ball in the 4 dimensional Euclidean space. Suppose that $u\in W^{2,2}(B)$, then $e^{u}\in L^{q}$ for any $q\geq1$.

As we know this is exactly the critical case of sobolev imbedding where we fail to get $L^{\infty}$ bounds. Any suggestion and help would be appreciated.

My idea of this that fails: Note $\int_{B} e^{pu}\leq \sum_{k=1}^{\infty}\frac{p^{k}}{k!}\int_{B} |u|^{k}$. By Sobolev imbedding we know that $W^{2,2}(B)\hookrightarrow L^{k}$ for any $k\geq 1$. So we know that $$\parallel u\parallel_{L^{k}(B)}\leq C_{k} \parallel u \parallel_{W^{2,2}} $$ We can then get : $\int_{B} e^{pu}\leq \sum_{k=1}^{\infty}\frac{p^{k}}{k!} (C_{k})^{k} \parallel u \parallel_{W^{2,2}}^{k}$. So that it suffices to control $C_{k}$. But based on my calculation, $C_{k} \sim k$, from which it seems this argument would fail.

After carefully thinking about what I need, I get the following.

Lemma: Suppose $u\in W^{2,2}(B(r_{0}))$ where $B_{r_{0}}$ is a ball with radius $r_{0}$ in the 4 dimensional Euclidean space, then $e^{u}\in L^{q}(B_{\frac{r_{0}}{2}})$ for any $q\geq 1$.

Proof: by multiplying a cut off function $\eta$ which is equal to 1 on $B_{\frac{r_{0}}{2}}$, we can apply the theorem 7.15 on Gilbert&Trudinger's book, i.e. Elliptic partial differential equations of second order (which can be also found on the link provided by Pedro Lauridsen Ribeiro in the comment ) to get $$ \int_{B_{r_{0}}}e^{(\frac{\eta u}{C_{1}||D^{1}(\eta |u|)||^{4}})^{\frac{4}{3}}}\leq C_{2}|B_{r_{0}}| $$ which then help control any $||e^{u}||_{L^{q}(B_{\frac{r_{0}}{2}})}$ for any $q\geq 1$.

I got stuck in the following lemma:

Lemma1: Let $B$ be the unit ball in the 4 dimensional Euclidean space. Suppose that $u\in W^{2,2}(B)$, then $e^{u}\in L^{q}$ for any $q\geq1$.

Actuall, I realized that the lemma1 follows from Trudger's theorem. https://en.wikipedia.org/wiki/Trudinger%27s_theorem

---------------------------------------------------------

Below is my little thoughts about this problem in the beginning and just want to keep it for myself.

My idea of this that fails: Note $\int_{B} e^{pu}\leq \sum_{k=1}^{\infty}\frac{p^{k}}{k!}\int_{B} |u|^{k}$. By Sobolev imbedding we know that $W^{2,2}(B)\hookrightarrow L^{k}$ for any $k\geq 1$. So we know that $$\parallel u\parallel_{L^{k}(B)}\leq C_{k} \parallel u \parallel_{W^{2,2}} $$ We can then get : $\int_{B} e^{pu}\leq \sum_{k=1}^{\infty}\frac{p^{k}}{k!} (C_{k})^{k} \parallel u \parallel_{W^{2,2}}^{k}$. So that it suffices to control $C_{k}$. But based on my calculation, $C_{k} \sim k$, from which it seems this argument would fail.

After carefully thinking about what I need, I get the following.

Lemma2: Suppose $u\in W^{2,2}(B(r_{0}))$ where $B_{r_{0}}$ is a ball with radius $r_{0}$ in the 4 dimensional Euclidean space, then $e^{u}\in L^{q}(B_{\frac{r_{0}}{2}})$ for any $q\geq 1$.

Proof: by multiplying a cut off function $\eta$ which is equal to 1 on $B_{\frac{r_{0}}{2}}$, we can apply the theorem 7.15 on Gilbert&Trudinger's book, i.e. Elliptic partial differential equations of second order (which can be also found on the link provided by Pedro Lauridsen Ribeiro in the comment ) to get $$ \int_{B_{r_{0}}}e^{(\frac{\eta u}{C_{1}||D^{1}(\eta |u|)||^{4}})^{\frac{4}{3}}}\leq C_{2}|B_{r_{0}}| $$ which then help control any $||e^{u}||_{L^{q}(B_{\frac{r_{0}}{2}})}$ for any $q\geq 1$.