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## Studia Mathematica

2003 | 156 | 3 | 243-260
Tytuł artykułu

### One-sided discrete square function

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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.
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Rocznik
Tom
Numer
Strony
243-260
Opis fizyczny
Daty
wydano
2003
Twórcy
autor
• Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain
autor