Critical case of Sobolev Embedding

Question

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.

Answer 1

As pointed in my comment above, what you seem to be looking for is Trudinger's inequality (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, 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.