One-sided discrete square function

A. de la Torre ; J. L. Torrea

Studia Mathematica, Tome 157 (2003), p. 243-260 / Harvested from The Polish Digital Mathematics Library

Résumé

Let f be a measurable function defined on ℝ. For each n ∈ ℤ we consider the average $Aₙf(x) = 2^{-n} ∫_{x}^{x+2ⁿ} f$. The square function is defined as $Sf(x) = (∑_{n=-∞}^{∞} |Aₙf(x) - A_{n-1}f(x)|²)^{1/2}$. The local version of this operator, namely the operator $S₁f(x) = (∑_{n=-∞}^{0} |Aₙf(x) - A_{n-1}f(x)|²)^{1/2}$, is of interest in ergodic theory and it has been extensively studied. In particular it has been proved [3] that it is of weak type (1,1), maps $L^{p}$ into itself (p > 1) and $L^{∞}$ into BMO. We prove that the operator S not only maps $L^{∞}$ into BMO but it also maps BMO into BMO. We also prove that the $L^{p}$ boundedness still holds if one replaces Lebesgue measure by a measure of the form w(x)dx if, and only if, the weight w belongs to the $A⁺_{p}$ class introduced by E. Sawyer [8]. Finally we prove that the one-sided Hardy-Littlewood maximal function maps BMO into itself.

Détail

Publié le : 2003-01-01

Zbl 1017.42009

EUDML-ID : urn:eudml:doc:286143