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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.

Let$X$be a rearrangement-invariant Banach function space on$R$^{$n$}and let$V$^{1}$X$be the Sobolev space of functions whose gradient belongs to$X$. We give necessary and sufficient conditions on$X$under which$V$^{1}$X$is continuously embedded into BMO or into$L$_{∞}. In particular, we show that$L$_{$n, ∞$}is the largest rearrangement-invariant space$X$such that$V$^{1}$X$is continuously embedded into BMO and, similarly,$L$_{$n$, 1}is the largest rearrangement-invariant space$X$such that$V$^{1}$X$is continuously embedded into$L$_{∞}. We further show that$V$^{1}$X$is a subset of VMO if and only if every function from$X$has an absolutely continuous norm in$L$_{$n, ∞$}. A compact inclusion of$V$^{1}$X$into$C$^{0}is characterized as well.

No abstract uploaded!

No abstract uploaded!