Bounded 𝐻^{∞}-calculus for the hydrostatic Stokes operator on 𝐿^{𝑝}-spaces and applications

Abstract

It is shown that the hydrostatic Stokes operator on L σ ¯ p ( Ω ) L^p_{\overline {\sigma }}(\Omega ) , where Ω ⊂ R 3 \Omega \subset \mathbb {R}^3 is a cylindrical domain subject to mixed periodic, Dirichlet and Neumann boundary conditions, admits a bounded H ∞ H^\infty -calculus on L σ ¯ p ( Ω ) L^p_{\overline {\sigma }}(\Omega ) for p ∈ ( 1 , ∞ ) p\in (1,\infty ) of H ∞ H^\infty -angle 0 0 . In particular, maximal L q − L p L^q-L^p -regularity estimates for the linearized primitive equations are obtained.