Triebel–Lizorkin–Lorentz Spaces and the Navier–Stokes Equations

Abstract

We derive basic properties of Triebel–Lizorkin–Lorentz spaces important in the treatment of PDE. For instance, we prove Triebel–Lizorkin–Lorentz spaces to be of class $\mathcal{H}\mathcal{T}$, to have property $(\alpha )$, and to admit a multiplier result of Mikhlin type. By utilizing these properties we prove the Laplace and the Stokes operator to admit a bounded $H^{\infty }$-calculus. This is finally applied to construct a unique maximal strong solution for the Navier–Stokes equations on corresponding Triebel–Lizorkin–Lorentz ground spaces.