Directional maximal operators and radial weights on the plane

Abstract

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 $$M_{\Omega,w}f(x) := \sup_{x\in R\in B_{\Omega}} \frac{1}{w(R)}\int_{R}|f(y)|w(y)\,dy,$$ where $B_{\Omega}$ denotes the all rectangles on the plane whose longest side is parallel to some unit vector in $\Omega$ and $w(R)$ denotes $\int_{R}w$. 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 weighted norm inequality $$\|M_{\Omega,w}f\|_{L^2(w)} \le C\log N \|f\|_{L^2(w)},$$ when $w(x)=|x|^a$, $a>0$, and when $\Omega$ is the set of unit vectors on the plane with cardinality $N$ sufficiently large. DOI: 10.1017/S0004972713000804