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 $S_{\mathcal {I}}$, originating from ergodic theory. First, we show that we can find some function $f\in L^\infty (\mathbb {R})$, such that $Sf$ equals infinity on a nonzero measurable set. Second, 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 $L^{\infty }_c$, the space of compactly supported $L^\infty (\mathbb {R})$ functions, to $BMO$ space. As a consequence, we solve an open question posed by Jones, Kaufman, Rosenblatt and Wierdl (2000). That is, $S_{\mathcal {I}}$ are uniformly bounded in $L^p(\mathbb {R})$ with respect to $\mathcal {I}$ for $2<p<\infty$.