On the R -boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer

In this paper, we prove the R-boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter λεΣ∈,γ0, where Σ∈,γ0={λεC||argλ|≤π-∈,|λ|≥γ0}(0<∈<πâ•2,γ0>0), and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0 < ∈ < π / 2 and γ<inf>0</inf> > 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large γ<inf>0</inf> > 0 for given 0 < ∈ < π / 2. We also prove the maximal L<inf>p</inf> - L<inf>q</inf> regularity theorem of the nonstationary Stokes problem as an application of the R-boundedness. The key of our approach is to apply several technical lemmas to the exact solution formulas of a resolvent problem. The formulas are obtained through the solutions of the ODEs, in the Fourier space, driven by the partial Fourier transform with respect to tangential space variable x′=(x1,...,xN-1).