One-sided discrete square function

A. de la Torre ¹   ; J. L. Torrea ²

¹ Departamento de Análisis Matemático Facultad de Ciencias Universidad de Málaga 29071 Málaga, Spain
² Departamento de Matemáticas Universidad Autónoma de Madrid 28049 Madrid, Spain

Studia Mathematica, Tome 156 (2003) no. 3, pp. 243-260

Résumé

Let $f$ be a measurable function defined on $\mathbb R$. For each $n\in{\mathbb Z}$ we consider the average $A_nf(x)=2^{-n}\int_x^{x+2^n}f$. The square function is defined as $$ Sf(x)=\Big(\sum_{n=-\infty}^\infty \vert A_nf(x)-A_{n-1}f(x)\vert^2\Big)^{1/2}. $$ The local version of this operator, namely the operator $$S_1f(x)= \Big(\sum_{n=-\infty}^0\vert A_nf(x)-A_{n-1}f(x)\vert^2\Big)^{1/2}, $$ is of interest in ergodic theory and it has been extensively studied. In particular it has been proved \cite{JKRW} that it is of weak type $(1,1)$, maps $L^p$ into itself ($p>1$) and $L^\infty$ into BMO. We prove that the operator $S$ not only maps $L^\infty$ 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 \cite{S}. Finally we prove that the one-sided Hardy–Littlewood maximal function maps BMO into itself.

Détail

DOI : 10.4064/sm156-3-3

Keywords: measurable function defined mathbb each mathbb consider average n int square function defined sum infty infty vert a n vert local version operator namely operator sum infty vert a n vert interest ergodic theory has extensively studied particular has proved cite jkrw weak type maps itself infty bmo prove operator only maps infty bmo maps bmo bmo prove boundedness still holds replaces lebesgue measure measure form only weight belongs class introduced nbsp sawyer nbsp cite finally prove one sided hardy littlewood maximal function maps bmo itself