The <Campanato space> $\mathcal L^{p,\mu}(\Omega)$ consists of $f\in L^p(\Omega)$ for which
$$
[f]_{\mathcal L^{p,\mu}}^p
=\sup_{x_0,r}r^{-\mu}\int_{\Omega\cap B_r(x_0)}
|f-f_{\Omega\cap B_r(x_0)}|^p<\infty,
$$
where $f_E=|E|^{-1}\int_Ef$. On a smooth bounded domain,
$$
\mathcal L^{p,n+p\alpha}(\Omega)=C^{0,\alpha}(\overline\Omega)
$$
with equivalent norms for $0<\alpha\leq1$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 107 / 3 / a / Solution

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{title2=$\mathcal L^{p,\mu}$}
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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.

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