Let $u:[0,1]\to\mathbb{R}^n$ be a bounded Borel function. It is well-known that if, for any compact interval $I\subseteq [0,1]$, $$ \int_I|u-u_I|^2\le C|I|^{1+\alpha} $$ for some $C,\alpha>0$ (here $u_I:=\frac{1}{|I|}\int_I u$), then $u$ is in fact $\frac{\alpha}{2}$-Holder continuous: this was first proved by Campanato in 1963.

**Q:** Is it true that if
$$ \int_I|u-u_I|^2\le |I|\omega(|I|) $$
for any $I$ then $u$ is continuous? Here $\omega$ denotes an arbitrary modulus of continuity. If this is false in general, can one characterize the $\omega$'s for which this is true?