Continuity and differentiability of Nemytskii operators on the Hardy space $$\mathcal{H}^{1,1} (T^1 )$$

John F. Toland Department of Mathematical Sciences, University of Bath

TBD mathscidoc:1701.332969

Arkiv for Matematik, 39, (2), 383-394, 2000.2
Let $$\mathcal{H}^{1,1} (T^1 )$$ denote the Hardy space of real-valued functions on the unit circle with weak derivatives in the usual real Hardy space $$\mathcal{H}^1 (T^1 )$$ . It is shown that when the weak derivative of a locally Lipschitz continuous function$f$has bounded variation on compact sets the Nemytskii operator$F$, defined by$F(u)=f·u$, maps $$\mathcal{H}^{1,1} (T^1 )$$ continuously into itself. A further condition sufficient for the continuous Fréchet differentiability of$F$is then added.
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@inproceedings{john2000continuity,
  title={Continuity and differentiability of Nemytskii operators on the Hardy space $$\mathcal{H}^{1,1} (T^1 )$$ },
  author={John F. Toland},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203603023154778},
  booktitle={Arkiv for Matematik},
  volume={39},
  number={2},
  pages={383-394},
  year={2000},
}
John F. Toland. Continuity and differentiability of Nemytskii operators on the Hardy space $$\mathcal{H}^{1,1} (T^1 )$$ . 2000. Vol. 39. In Arkiv for Matematik. pp.383-394. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203603023154778.
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