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. 



Critical case of Sobolev Embedding
As pointed in my comment above, what you seem to be looking for is [Trudinger's inequality](https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Orlicz_space#Relations_to_Sobolev_spaces) (see e.g. formula (7.40), pp. 162 of the book *Elliptic Partial Differential Equations of Second Order* by D. Gilbarg and N.S. Trudinger (Springer-Verlag, 1998)), with the proviso that in fact you need $u\in W^{2,2}\_0(B)=$ closure of $\mathscr{C}^\infty\_c(B)$ in $W^{2,2}(B)$. Since $B$ has Lipschitz boundary, Trudinger's inequality states that for all $u\in W^{2,2}\_0(B)$ there are constants $C\_1,C\_2>0$ such that $$\tag{1}\label{e1}\int\_B\exp\left(\left(\frac{|u(x)|}{C\_1\|D^2 u\|\_2}\right)^2\right)\mathrm{d}x\leq C\_2|B|\ ,$$ where $|B|$ stands for the Lebesgue measure of $B$. Notice that if $u$ is constant on $B$ then $\|D^2 u\|\_2=0$ and in this case the left hand side of the above inequality equals $+\infty$, hence the need for taking $u\in W^{2,2}\_0(B)$.


It is not difficult to infer from inequality \eqref{e1} that $e^u\in L^q(B)$ for all $q>0$. To that end, set $a=C\_1\|D^2 u\|\_2$ ($>0$!) and notice that whenever $|u(x)|\geq a^2 q$ we have that $e^{q|u(x)|}\leq e^{(\frac{|u(x)|}{a})^2}$ and whenever $|u(x)|\leq a^2 q$ we have that $e^{q|u(x)|}\leq e^{a^2q^2}$. Therefore $$\int\_B |e^{u(x)}|^q\mathrm{d}x\leq\int\_B e^{q|u(x)|}\mathrm{d}x\leq e^{a^2q^2}|B|+\int\_B e^{(\frac{|u(x)|}{a})^2}\mathrm{d}x\leq (e^{a^2q^2}+C\_2)|B|<\infty\ ,$$ as desired.


Trudinger's inequality can indeed be seen as a sharpening of Sobolev's embedding theorem. Interestingly, Trudinger's original argument ([*On Imbeddings into Orlicz Spaces and Some Applications*](http://www.iumj.indiana.edu/oai/1968/17/17028/17028.xml), J. Math. Mech. **17** (1967) 473-483) is not far off what you were trying to do - it consisted in controlling each individual term in the power series for the exponential but that involved estimating the action of certain Riesz potentials.


