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The Fefferman–Stein Type Inequalities for Strong and Directional Maximal Operators in the Plane
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- Journal:
- Canadian Mathematical Bulletin / Volume 61 / Issue 1 / 01 March 2018
- Published online by Cambridge University Press:
- 20 November 2018, pp. 191-200
- Print publication:
- 01 March 2018
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- Article
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DIRECTIONAL MAXIMAL OPERATORS AND RADIAL WEIGHTS ON THE PLANE
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 89 / Issue 3 / June 2014
- Published online by Cambridge University Press:
- 12 September 2013, pp. 397-414
- Print publication:
- June 2014
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- Article
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- You have access
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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 where
$$\begin{eqnarray*}{M}_{\Omega , w} f(x): = \sup _{x\in R\in \mathcal{B} _{\Omega }}\frac{1}{w(R)} \int \nolimits \nolimits_{R} \vert f(y)\vert w(y)\hspace{0.167em} dy,\end{eqnarray*}$$
${ \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 when
$$\begin{eqnarray*}\Vert {M}_{\Omega , w} f\mathop{\Vert }\nolimits_{{L}^{2} (w)} \leq C\log N\Vert f\mathop{\Vert }\nolimits_{{L}^{2} (w)} ,\end{eqnarray*}$$
$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.
