Campanato space (${\mathcal{L}}^{p,\mu }$)

The Campanato seminorm measures mean oscillation byOn regular bounded domains, ${\mathcal{L}}^{p,n+p\alpha }=C^{0,\alpha }$ for $0<\alpha \le 1$, with equivalent norms.

If a nondecreasing oscillation quantity satisfiesfor $0<\rho \le R$, with $\gamma >\beta$, then iteration gives $\Phi (\rho )\le C(\rho /R{)}^{\beta }\Phi (R)+CB{\rho }^{\beta }$.