For strong solutions of the incompressible Navier-Stokes
equations in bounded domains with velocity specified at the boundary, we
proof unconditional stability and obtain error estimates of discretization
schemes that decouple the updates of pressure and velocity through
explicit time-stepping for pressure. The proofs are simple, based upon a
new, sharp estimate for the commutator of the Laplacian and Helmholtz
projection operators. This allows us to treat an unconstrained formulation
of the Navier-Stokes equations as a perturbed diffusion equation.