  This paper is concerned with the global well-posedness of the two-dimensional
incompressible vorticity equation in the half plane. Under the assumption that
the initial vorticity $\omega_0\in W^{k,p}(\R^{2}_+)$ with $k\geq3$ and
$1<p<2$, it is shown that the two-dimensional incompressible vorticity equation
admits a unique solution $\omega\in C([0,T];W^{k,p}(\R^{2}_+))$ for any $T>0$.
An elementary and self-contained proof is presented and delicate estimates of
the velocity and its derivatives are obtained in this paper. It should be
emphasized that the uniform estimate on
$\int^t_0\|u(\tau)\|_{W^{1,\infty}(\R^2_+)}d\tau$ is required to complete the
global regularity of the solution. To do that, the double exponential growth in
time of the gradient of the vorticity in the half plane is established and
applied. This is different from the proof of global well-posedness of the Euler
velocity equations in the Sobolev spaces, in which a Kato-type or
logarithmic-type estimate of the gradient of the velocity is enough to close
the energy estimates.
