= Morrey's inequality
{c}
{title2=$|u(x)-u(y)|\leq C_{d,p}|x-y|^{1-d/p}\|\nabla u\|_{L^p}$}
= Morrey inequality
{c}
{synonym}
For $d<p<\infty$, every <Sobolev space> element $u\in W^{1,p}(\mathbb R^d)$ has a <Hölder continuous function> representative of exponent $1-d/p$, with the displayed bound. It also satisfies $\|u\|_\infty\leq C_{d,p}(\|u\|_p+\|\nabla u\|_p)$. Averaging the <fundamental theorem of calculus along a line segment> over a ball gives
$$
|u(x)-u_{B(x,r)}|\leq C_d\int_{B(x,r)}\frac{|\nabla u(z)|}{|x-z|^{d-1}}\,dz.
$$
The <Holder inequality> bounds this by $C_{d,p}r^{1-d/p}\|\nabla u\|_p$, because $(d-1)p'<d$. To compare two ball averages, translate a ball along the segment between their centres and use the same <fundamental theorem of calculus along a line segment>. These bounds give the displayed estimate. <Density of smooth functions in a Sobolev space> supplies the representative for nonsmooth $u$. For $p=\infty$, the corresponding conclusion is <Lipschitz continuity>.
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