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

TL;DR

This paper investigates multiplication and Nemytskij operators in anisotropic vector-valued function spaces, providing optimal estimates and applications to improve results on quasilinear evolution equations.

Contribution

It generalizes and improves known estimates for multiplication and Nemytskij operators in anisotropic vector-valued Besov, Bessel potential, and Sobolev-Slobodeckij spaces, with applications to PDEs.

Findings

Optimal estimates for multiplication in anisotropic spaces.

Extension of Nemytskij operator theory to anisotropic vector-valued spaces.

Enhanced results on quasilinear evolution equations.

Abstract

We study multiplication as well as Nemytskij operators in anisotropic vector-valued Besov spaces $B^{s, \omega}_p$, Bessel potential spaces $H^{s, \omega}_p$, and Sobolev-Slobodeckij spaces $W^{s, \omega}_p$. 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.