= Solution
The <planar vorticity velocity kernel> has magnitude $(2\pi|z|)^{-1}$. Split its defining <integral> into $|x-y|<R$ and $|x-y|\geq R$ for any $R>0$. The singular part is absolutely integrable in two dimensions:
$$
|U[\Omega](x)|
\leq \frac{\|\Omega\|_\infty}{2\pi}
\int_{|z|<R}\frac{dz}{|z|}
+\frac{\|\Omega\|_1}{2\pi R}
=R\|\Omega\|_\infty+\frac{\|\Omega\|_1}{2\pi R}.
$$
In particular $R=1$ proves
$$
\boxed{\|U[\Omega]\|_\infty
\leq\|\Omega\|_\infty+\frac1{2\pi}\|\Omega\|_1.}
$$
When both norms are nonzero, optimization in $R$ also gives $\|U[\Omega]\|_\infty\leq\sqrt{2/\pi}\,\|\Omega\|_1^{1/2}\|\Omega\|_\infty^{1/2}$. If either norm is zero, $\Omega=0$ almost everywhere. Absolute convergence supplies a well-defined velocity at every $x$, with these uniform bounds.
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