It is an important problem to find the time-dependent estimate of the Navier-Stokes equations on the exterior domain of half-space. The Stokes equations is a linearized Navier-Stokes equations where the nonlinear convective term of the stationary Navier-stokes equations is removed.
If the analyticity of the Stokes operators is shown on the half-space, then the semigroup operator theory gives the time dependent estimate of the nonstationary Stokes equation. And the weight may be helpful to find the sharp estimate.
In this paper I tried to show the analiticity of Stokes operator in a weighted space $L^p_{\gamma}(R^3_+)$.
This paper also showed that the Hodge decomposition holds on the weighted space $L^p(R^3_+)$, and showed the Stein's multiplier theorem which has a little improvement of the older one. But this paper is not complete for the main part of the theorem does not have proved yet.
