Lemma 2.9.16 (Morrey's inequality)DE7722E667E2

Proof. We prove the estimate for smooth $v$, pass to general $v$ by mollification, and then apply the result to first derivatives. Assume first that $v$ is smooth on $B$. Let $Q\subset B$ be a ball of radius $r$, let $v_ Q=|Q|^{-1}\int _ Qv$ be the average of $v$ over $Q$, and let $x\in Q$. We claim that

For $z\in Q$, write $z=x+s\theta $ with $s=|z-x|$ and $\theta \in S^{d-1}$; then $|v(z)-v(x)|\leq \int _0^ s|\nabla v(x+t\theta )|\, \mathrm d t$, and $|v(x)-v_ Q|$ is at most the average over $z\in Q$ of these segment integrals. Since $Q$ is convex and open and contains $x$, for each $\theta $ the radii $s>0$ with $x+s\theta \in Q$ form an interval $0<s<R(\theta )$, and $R(\theta )\leq 2r$. In polar coordinates about $x$ ((3.1.2) on $B(x,2r)\supset Q$), the average of the segment integrals is

Interchanging the integrals in $s$ and $t$ produces the factor $|Q|^{-1}\int _ t^{R(\theta )}s^{d-1}\, \mathrm d s$, which is at most $(2r)^ d/(d|Q|)$ and hence bounded by a constant depending only on $d$. The remaining integral $\int _{S^{d-1}}\int _0^{R(\theta )}|\nabla v(x+t\theta )|\, \mathrm d t\, \mathrm d S(\theta )$ is the integral on the right side of the claim, written in polar coordinates. This proves the claim.

By Hölder's inequality with $p'=p/(p-1)$, the right side of the claim is at most $C_{d,p}r^{1-d/p}\| \nabla v\| _{L^ p(Q)}$, because

and this integral is finite exactly when $(1-d)p'>-d$, that is, when $p>d$. (For $d=1$ the exponent $(1-d)p'$ is zero.)

Let $\delta >0$ be the distance from $B'$ to $\mathbb R^ d\setminus B$. If $x,y\in B'$ and $0<|x-y|<\delta /2$, the ball $Q=B(x,2|x-y|)$ lies in $B$ and contains $x$ and $y$. Applying the preceding bound at both points gives

For $x\in B'$, the same bound on the ball $Q=B(x,\delta /2)\subset B$, together with the estimate $|v_ Q|\leq |Q|^{-1/p}\| v\| _{L^ p(B)}$ from Hölder's inequality, gives $\sup _{B'}|v|\leq C\| v\| _{W^{1,p}(B)}$ with $C$ depending on $d,p,\delta $. If $|x-y|\geq \delta /2$, then $|v(x)-v(y)|/|x-y|^\alpha \leq 2(\delta /2)^{-\alpha }\sup _{B'}|v|$. These bounds prove the estimate for smooth $v$ on $B'$, and hence on $\overline{B'}$, since $v$ is continuous on $B\supset \overline{B'}$.

For general $v\in W^{1,p}(B)$, choose a ball $B''$, depending only on $B'$ and $B$, with $B'\Subset B''\Subset B$, and let $v_\varepsilon =v*\rho _\varepsilon $, which is smooth on $B''$ for small $\varepsilon $. By the remark preceding lemma 2.9.14, $\partial _ av_\varepsilon =(\partial _ av)_\varepsilon $, and $v_\varepsilon \to v$ and $\partial _ av_\varepsilon \to \partial _ av$ in $L^ p(B'')$. The estimate for smooth functions, applied on $B'\Subset B''$ to the differences of the $v_\varepsilon $, shows that they form a Cauchy family in the norm of $C^{0,\alpha }(\overline{B'})$. Their uniform limit is continuous on $\overline{B'}$, agrees almost everywhere with $v$, and satisfies the asserted estimate, because $\| v\| _{W^{1,p}(B'')}\leq \| v\| _{W^{1,p}(B)}$. Two continuous functions that agree almost everywhere on an open set agree everywhere on it. Hence the continuous representatives obtained from different balls $B'\Subset B$ agree on their overlaps and define a single continuous representative on $B$.

Finally, let $v\in W^{2,p}_{\mathrm{loc}}(U)$. On every ball $B\Subset U$, both $v$ and its first derivatives $\partial _ av$ belong to $W^{1,p}(B)$. By the preceding paragraph, on each smaller ball $B'\Subset B$ the mollifications $v_\varepsilon $ and $\partial _ av_\varepsilon =(\partial _ av)_\varepsilon $ converge uniformly to the continuous representatives $\tilde v$ of $v$ and $\tilde w_ a$ of $\partial _ av$. Passing to the limit in

shows that $\tilde v$ has the continuous partial derivatives $\partial _ a\tilde v=\tilde w_ a$. Hence $\tilde v$ is continuously differentiable, and by the first part its first derivatives are Hölder continuous with exponent $\alpha $ on each $\overline{B'}$. Every compact subset of $U$ is covered by finitely many such balls $B'$; pairs of points closer than a Lebesgue number of this cover lie in a common ball, and for the remaining pairs the supremum bound suffices. Hence $\tilde v\in C^{1,\alpha }_{\mathrm{loc}}(U)$. $\square$