Morrey's inequality

Remarks

The key step is the bound $\frac{1}{\abs{B}} \int_B \abs{u(y) - u(x)} \, dy \le C \int_B \frac{\abs{\nabla u(y)}}{\abs{x - y}^{n - 1}} \, dy$ for a ball $B$ centred at $x$. By Hölder's inequality, the kernel $\abs{x - y}^{1 - n}$ lies in $L^{p'}(B)$ exactly when $p > n$, and this gives a bound by $r^{1 - n/p} \norm{\nabla u}_{L^p}$ for the ball of radius $r$. Comparing $u(x)$ and $u(y)$ with the average of $u$ on a ball containing both gives the Hölder continuity bound.

The hypothesis $p > n$ is needed. For $n = 2$ and $p = 2$, the function $\log \log (1/\abs{x})$, cut off away from the origin, lies in $W^{1,2}$ but is unbounded near $0$.

For a bounded open set $U$ with $C^1$ boundary, the same holds with $\R^n$ replaced by $U$ and $C$ depending also on $U$. Together with the Sobolev inequality it forms the Sobolev embedding theorem.

Going deeper

For a student

Morrey's inequality is how one shows that weak solutions are continuous functions. In dimension $n$, a function in $W^{k,p}$ with $kp > n$ is continuous, by combining Morrey's inequality with the Sobolev inequality for the lower derivatives. This is the step that turns $L^2$ regularity results, such as those of elliptic regularity, into classical smoothness.