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DIRECTIONAL MAXIMAL OPERATORS AND RADIAL WEIGHTS ON THE PLANE

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  12 September 2013

HIROKI SAITO  and
HITOSHI TANAKA
HIROKI SAITO*
Affiliation:
Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji-shi, Tokyo 192-0397, Japan
HITOSHI TANAKA
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, Tokyo 153-8914, Japan email [email protected]
*
✉[email protected]
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Abstract

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Let $\Omega $ be the set of unit vectors and $w$ be a radial weight on the plane. We consider the weighted directional maximal operator defined by

$$\begin{eqnarray*}{M}_{\Omega , w} f(x): = \sup _{x\in R\in \mathcal{B} _{\Omega }}\frac{1}{w(R)} \int \nolimits \nolimits_{R} \vert f(y)\vert w(y)\hspace{0.167em} dy,\end{eqnarray*}$$
where ${ \mathcal{B} }_{\Omega } $ denotes the set of all rectangles on the plane whose longest side is parallel to some unit vector in $\Omega $ and $w(R)$ denotes $\int \nolimits \nolimits_{R} w$. In this paper we prove an almost-orthogonality principle for this maximal operator under certain conditions on the weight. The condition allows us to get the weighted norm inequality
$$\begin{eqnarray*}\Vert {M}_{\Omega , w} f\mathop{\Vert }\nolimits_{{L}^{2} (w)} \leq C\log N\Vert f\mathop{\Vert }\nolimits_{{L}^{2} (w)} ,\end{eqnarray*}$$
 when $w(x)= \vert x\hspace{-1.2pt}\mathop{\vert }\nolimits ^{a} $, $a\gt 0$, and when $\Omega $ is the set of unit vectors on the plane with cardinality $N$ sufficiently large.

Keywords

almost-orthogonality principledirectional maximal operatorradial weightstrong-type estimate

MSC classification

Secondary: 42B25: Maximal functions, Littlewood-Paley theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 3 , June 2014 , pp. 397 - 414
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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