  We study the Cauchy problem for the quasi-geostrophic equations with the
critical dissipation in the two dimensional half space under the homogeneous
Dirichlet boundary condition. We show the global existence, the uniqueness and
the analyticity of solutions, and the real analyticity up to the boundary is
obtained. We will show one of natural ways to estimate the nonlinear term for
functions satisfying the Dirichlet boundary condition.
