Planar vorticity velocity kernel ($K(z)=\pm z^{\perp }/(2\pi |z{|}^2)$)

The velocity induced by planar vorticity is a convolution with this kernel, whose sign depends on the stream function and vorticity conventions. Its magnitude is $(2\pi |z|{)}^{-1}$ and its derivative is bounded by $C|z{|}^{-2}$. Splitting the convolution at radius $R$ gives $\Vert K\ast \omega {\Vert }_{\infty }\le R\Vert \omega {\Vert }_{\infty }+\Vert \omega {\Vert }_1/(2\pi R)$. The same singularity yields a log-Lipschitz modulus, enough for a Yudovich characteristic flow.