Modulus of continuity ($\omega (f,\delta )$)

The modulus of continuity of a function $f$ isIt quantifies uniform continuity by recording the largest oscillation of $f$ across pairs of points at distance at most $\delta$.

Set $\mu (0)=0$ and extend $\mu (r)=1$ for $r\ge 1$. This positive, nondecreasing modulus of continuity is weaker than a linear Lipschitz continuity bound but still has ${\int }_0^{1}dr/\mu (r)=\infty$. At small distances it is equivalent to $r\log (1/r)$. A signed $r\log r$ cannot be an upper modulus at distances less than one. The planar vorticity velocity kernel maps $L^1\cap L^{\infty }$ into velocity fields with this modulus.