# MR2819034 Castillo, René Erlín The Nemytskii operator on bounded p-variation in the mean spaces. Mat. Enseñ. Univ. (N. S.) 19 (2011), no. 1, 31–41. (Reviewer: Pasquale Vetro)

Risultato della ricerca: Review article

### Abstract

The author introduces the notion of bounded $p$-variation in thesense of $L_p$-norm. Precisely: Let $f \in L_p[0,2\pi]$ with$1<p<\infty$. Let $P: 0=t_0 <t_1< \cdots <t_n=2\pi$ be a partion of$[0,2\pi]$ if $$V_p^m(f,T) = \sup \{\sum_{k=1}^{n}\int_T\frac{|f(x+t_k)-f(x+t_{k-1})|^p)}{|t_k-t_{k-1}|^{p-1}}\}<\infty,$$ where the supremum is taken over all partitions $P$ of$[0,2\pi]$ and $T=\mathbb{R}/2\pi \mathbb{Z}$, then $f$ is said tobe of bounded $p$-variation in the mean. The author obtains a Riesztype result for functions of bounded $p$-variation in the mean andgives some properties for functions of bounded $p$-variation byusing the Nemytskii operator.
Lingua originale English 0 MATHEMATICAL REVIEWS 2012 Published - 2012