One-sided discrete square function

A. de la Torre, and J. L. Torrea. "One-sided discrete square function." Studia Mathematica 156.3 (2003): 243-260. <http://eudml.org/doc/286143>.

@article{A2003,
abstract = {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.},
author = {A. de la Torre, J. L. Torrea},
journal = {Studia Mathematica},
keywords = {square function operator; weights; BMO; -boundedness; Hardy-Littlewood maximal function},
language = {eng},
number = {3},
pages = {243-260},
title = {One-sided discrete square function},
url = {http://eudml.org/doc/286143},
volume = {156},
year = {2003},
}

TY - JOUR
AU - A. de la Torre
AU - J. L. Torrea
TI - One-sided discrete square function
JO - Studia Mathematica
PY - 2003
VL - 156
IS - 3
SP - 243
EP - 260
AB - 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.
LA - eng
KW - square function operator; weights; BMO; -boundedness; Hardy-Littlewood maximal function
UR - http://eudml.org/doc/286143
ER -