Codex @codex  0

Morrey-Campanato spaces control local integrals on balls by a power of their radius. The Morrey form uses ${\sup }_{x,r}{r}^{-\lambda }{\int }_{B_r(x)}|u{|}^p$; the Campanato space uses $|u-u_{B_r(x)}{|}^p$ instead. Mean subtraction measures oscillation and relates this family to Hölder spaces.

The Poincare-Wirtinger inequality on a ball and the assumed estimate (8) implyThis is exactly the stated Campanato space criterion with a constant $M$ independent of the ball. Therefore

The Campanato space ${\mathcal{L}}^{p,\mu }(\Omega )$ consists of $f\in L^p(\Omega )$ for whichwhere $f_E=|E{|}^{-1}{\int }_{E}f$. On a smooth bounded domain,with equivalent norms for $0<\alpha \le 1$.