The behaviour of square functions from ergodic theory in L^∞

2014/10/06 by Guixiang Hong, Hong, Guixiang
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 42B25 #Secondary 47G10 #math.CA #msc:42B25 #msc:47G10

paper · pdf · doi:10.48550/arxiv.1410.1574

arxiv created 2014/10/06 · arxiv updated 2014/10/08

Abstract

In this paper, we analyze carefully the behaviour in L^∞(\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∈ L^∞(ℝ), such that Sf equals infinity on a nonzero measure set. Secondly, we can find compact supported function f∈ L^∞(ℝ) and \mathcal I such that S_I f does not belong to BMO space. Finally, we show that S is bounded from L^∞_c to BMO space. As a consequence, we solve an open question posed by Jones, Kaufman, Rosenblatt and Wierdl in \citeJKRW98. That is, S_\mathcal I are uniformly bounded in L^p(\mathbb R) with respect to \mathcal I for 2<p<∞.

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