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Multiplication in vector‐valued anisotropic function spaces and applications to non‐linear partial differential equations

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  • Matthias Köhne
  • Jürgen Saal

Abstract

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

Suggested Citation

  • Matthias Köhne & Jürgen Saal, 2022. "Multiplication in vector‐valued anisotropic function spaces and applications to non‐linear partial differential equations," Mathematische Nachrichten, Wiley Blackwell, vol. 295(9), pages 1709-1754, September.
    • Handle: RePEc:bla:mathna:v:295:y:2022:i:9:p:1709-1754
    DOI: 10.1002/mana.202000111
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