Campanato space
= Campanato space
{c}
{title2=$\mathcal L^{p,\mu}$}
{wiki=Campanato_space}
The Campanato seminorm measures mean oscillation by
$$
[f]_{\mathcal L^{p,\mu}(\Omega)}^p
=\sup_{x,r}r^{-\mu}\int_{\Omega\cap B_r(x)}|f-f_{\Omega\cap B_r(x)}|^p.
$$
On regular bounded domains, $\mathcal L^{p,n+p\alpha}=C^{0,\alpha}$ for $0<\alpha\leq1$, with equivalent norms.