Evans shows a general embedding of Sobolev Spaces of bounded open sets with regular boundary onto holder continuous functions.
My question is from a generalization of this result to $\mathbb{R}^n$, following the proof step by step works except for some critical cases the cases $W^{k,p}(R^n)$ for $p$ a multiple of $n$. My question is:
Is the space $W^{2,2}(\mathbb{R}^2)$ embedded in holder continuous functions for some positive exponent? If so, how can one begin to show this?
As stated in the comments, one has the embedding $$ W^{2,2}(\mathbb R^2) \hookrightarrow C^{0,\alpha}(\mathbb R^2) $$ for all $\alpha \in (0,1)$. However, unlike in the non-critical spaces, you do not get a homogeneous estimate, that is $$ [ u ]_{C^{0,\alpha}(\mathbb R^2)} \not\leq C \lVert \mathrm{D}^2 u\rVert_{L^2(\mathbb R^2)} $$ for any $\alpha \in (0,1)$, but rather just a estimate of the full norms $$ \lVert u \rVert_{C^{0,\alpha}(\mathbb R^2)} \leq C \lVert u \rVert_{W^{2,2}(\mathbb R^2)}.$$ Let $u \in W^{2,2}(\mathbb R^2)$, and applying the general Sobolev inequality (Theorem 6) locally we can assume that $u$ is continuous. The idea is to interpolate between the $L^2$ norms of $u$ and $\mathrm{D}^2u$, following the second step of the proof of Theorem 4 (Morrey's inequality). That is in equation (22) of said proof, one establishes the inequality $$ \lvert u(x) \rvert \leq C \left(\int_{B(x,1)} \lvert \mathrm{D} u \rvert^p \,\mathrm{d}y\right)^{\frac1p} \left( \int_{B(x,1)} \lvert x - y \rvert^{-(n-1) \frac{p}{p-1}} \,\mathrm{d} y\right)^{\frac{p-1}p} + \lVert u \rVert_{L^2(\mathbb R^2)} $$ for any fixed $p>2$. Note that this is slightly different from what is stated in the book, but it follows from the same reasoning. Now we need to deal with the gradient term, for which we use Sobolev embedding to estimate $$ \lVert \mathrm{D} u \rVert_{L^p(B(x,1))} \leq C\lVert u \rVert_{W^{2,2}(\mathbb R^2)},$$ where the constant $C$ is independent of $x$. Here I've appealed again to Theorem 6, noting it is always applied on a ball of radius 1. Hence combining these estimates gives $$ \lvert u(x) \rvert \leq C \lVert u \rVert_{W^{2,2}(\mathbb R^2)} $$ for all $x \in \mathbb R^2$, establishing a supremum bound.
The above relied on an interpolation argument, where we use the $W^{2,2}$-regularity in the small ball $B(x,1)$, and outside it we use the $L^2$-boundedness of $u$. To argument to upgrade this to Hölder continuity is similar, where we fix $x \in \mathbb R^2$. Then by Theorem 6 we know that if $y \in \mathbb R^2$ such that $y \neq x$ and $\lvert y - x \rvert \leq 1$, we have $$ \frac{\lvert u(x) - u(y)\rvert}{\lvert x - y\rvert^{\alpha}} \leq [ u]_{C^{0,\alpha}(B(x,1))} \leq C\lVert u \rVert_{W^{2,2}(B(x,1))}. $$ Now if $\lvert y - x \rvert > 1$, then we simply use the supremum bound $$ \frac{\lvert u(x) - u(y)\rvert}{\lvert x - y\rvert^{\alpha}} \leq \lvert u(x) \rvert + \lvert u(y) \rvert \leq 2\lVert u \rVert_{L^{\infty}(\mathbb R^2)} \leq C \lVert u \rVert_{W^{2,2}(\mathbb R^2)}.$$ Hence combining these two estimates, and noting that $x$ was arbitrary, we deduce that $$ [u]_{C^{0,\alpha}(\mathbb R^2)} = \sup_{x \neq y} \frac{\lvert u(x) - u(y)\rvert}{\lvert x - y\rvert^{\alpha}} \leq C \lVert u \rVert_{W^{2,2}(\mathbb R^2)}.$$ Hence combined with the supremum bound we have established the claimed estimate, and thus the desired embedding.
This type of argument does apply more generality in these endpoint cases, which is an instructive exercise to verify.
Finally note that we cannot take $\alpha = 1$, as the Sobolev embedding does fail at this endpoint scale. See Exercise 13 at the end of the chapter for a related counterexample (which shows that $W^{1,2} \not\hookrightarrow L^{\infty}$), and in general identifying the correct spaces one embeds to at the endpoints is more delicate.