Multiplication in Vector-Valued Anisotropic Function Spaces and Applications to Non-Linear Partial Differential Equations

We study multiplication as well as Nemytskij operators in anisotropic vector-valued Besov spaces $B_p^{s,\omega }$, Bessel potential spaces $H^{s, \omega}_p$,andSobolev-Slobodeckijspaces$W^{s, \omega}_p$. Concerning multiplication we obtain optimal estimates, which constitute generalizations and improvements of known estimates in the isotropic/scalar-valued case. Concerning Nemytskij operators we consider the acting of analytic functions on supercritial anisotropic vector-valued function spaces of the above type. Moreover, we show how the given estimates may be used in order to improve results on quasilinear evolution equations as well as their proofs.