The behaviour of square functions from ergodic theory in $L^{\infty }$

Abstract

In this paper, we analyze carefully the behaviour in $L^{\infty }(\mathbb{R})$ of the square functions $S$ and $S_{\mathcal{I}}$'s, originating from ergodic theory. Firstly, we show that we can find some function $f\in L^{\infty }(\mathbb{R})$, such that $Sf$ equals infinity on a nonzero measure set. Secondly, we can find compact supported function $f\in L^{\infty }(\mathbb{R})$ and $\mathcal{I}$ such that $S_{\mathcal{I}}f$ does not belong to $BMO$ space. Finally, we show that $S$ is bounded from $L_c^{\infty }$ to $BMO$ space. As a consequence, we solve an open question posed by Jones, Kaufman, Rosenblatt and Wierdl in \cite{JKRW98}. That is, $S_{\mathcal{I}}$ are uniformly bounded in $L^p(\mathbb{R})$ with respect to $\mathcal{I}$ for $2<p<\infty$.

Cited by in corpus (1)

• Vector valued $q$-variation for differential operators and semigroups I