## Right Type Departmental Bulletin Paper

### Abstract

A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function HITOSHI TANAKA* () For $p&gt;1$ 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
OAI identifier: oai:CiteSeerX.psu:10.1.1.909.577
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