One-sided discrete square function
Tom 156 / 2003
Streszczenie
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.
