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By Littlewood Maximal Function (harmonic Analysis


A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function HITOSHI TANAKA* () For $p>1 $ and $d\geq 1 $ J. Kinnunen proved that if $f $ is afunction on the Sobolev space $W^{1,p}(\mathrm{R}^{d}) $ , then the Hardy-Littlewood maximal function $\mathcal{M}f $ has the first order weak partial derivatives which belong to $L^{\mathrm{p}}(\mathrm{R}^{d}) $ and whose $IP$-norm are controlled by those of $f $. We improve Kinnunen’s result to $p=1 $ and $d=1 $ by making an expression of the maximal function by the one-sided maximal functions. We also study some properties of the one-sided maximal functions. 1Introduction For $f $ alocally integrable function on $\mathrm{R}^{d} $ , $d\geq 1 $ , define the Hardy-Littlewood maximal function $\mathcal{M}f $ as $ ( \mathcal{M}f)(x)=\sup_{Q}\frac{1}{|Q|}\int_{Q}|f|dy $, where the supremum is taken over all cubes $Q $ containing $x\in \mathrm{R}^{d} $. Here, $|Q| $ denote

Year: 2016
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