  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 $\mathcal{HT}$, 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 derive local strong well-posedness for the Navier-Stokes
equations on corresponding Triebel-Lizorkin-Lorentz ground spaces.
