  We revisit Yudovich's well-posedness result for the $2$-dimensional Euler
equations for an inviscid incompressible fluid on either a sufficiently regular
(not necessarily bounded) open set $\Omega\subset\mathbb{R}^2$ or on the torus
$\Omega=\mathbb{T}^2$.
  We construct global-in-time weak solutions with vorticity in $L^1\cap
L^p_{\mathrm{ul}}$ and in $L^1\cap Y^\Theta_{\mathrm{ul}}$, where
$L^p_{\mathrm{ul}}$ and $Y^\Theta_{\mathrm{ul}}$ are suitable
uniformly-localized versions of the Lebesgue space $L^p$ and of the Yudovich
space $Y^\Theta$ respectively, with no condition at infinity for the growth
function $\Theta$. We also provide an explicit modulus of continuity for the
velocity depending on the growth function $\Theta$. We prove uniqueness of weak
solutions in $L^1\cap Y^\Theta_{\mathrm{ul}}$ under the assumption that
$\Theta$ grows moderately at infinity. In contrast to Yudovich's energy method,
we employ a Lagrangian strategy to show uniqueness.
  Our entire argument relies on elementary real-variable techniques, with no
use of either Sobolev spaces, Calder\'on-Zygmund theory or Littlewood-Paley
decomposition, and actually applies not only to the Biot-Savart law, but also
to more general operators whose kernels obey some natural structural
assumptions.
