Morrey inequality on a cube ($|v(y)-v_Q|\le C{r}^{1-n/p}\Vert Dv{\Vert }_{L^p(Q)}$)

For a cube $Q$ of side length $r$ in ${\mathbb{R}}^n$, $p>n$, and a continuously differentiable function $v$, its average $v_Q=|Q{|}^{-1}{\int }_{Q}v$ satisfiesAveraging the fundamental theorem of calculus along a line segment from $y$ to $z\in Q$ and changing variables gives $|v(y)-v_Q|\le C_n{\int }_Q|Dv(w)||w-y{|}^{1-n}dw$. The Holder inequality applies because $(n-1)p^{\prime }<n$, and the kernel norm is $O(r^{1-n/p})$. Approximation extends the estimate to the continuous representative of a Sobolev space function. On ${\mathbb{R}}^n$ it yields a Hölder continuous function of exponent $1-n/p$ representing every $W^{1,p}$ class.