= Morrey inequality on a cube
{c}
{title2=$|v(y)-v_Q|\leq Cr^{1-n/p}\|Dv\|_{L^p(Q)}$}
For a cube $Q$ of side length $r$ in $\mathbb R^n$, $p>n$, and a <continuously differentiable function> $v$, its average $v_Q=|Q|^{-1}\int_Qv$ satisfies
$$
|v(y)-v_Q|\leq C_{n,p}r^{1-n/p}\|Dv\|_{L^p(Q)}\qquad(y\in Q).
$$
Averaging the <fundamental theorem of calculus along a line segment> from $y$ to $z\in Q$ and changing variables gives $|v(y)-v_Q|\leq C_n\int_Q|Dv(w)|\,|w-y|^{1-n}dw$. The <Holder inequality> applies because $(n-1)p'<n$, 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.
Back to article page
