Directional maximal operators and radial weights on the plane

Authors

  • H. Saito Tokyo Metropolitan University
  • H. Tanaka The University of Tokyo

Keywords:

almost-orthogonality principle, directional maximal operator, radial weight, strong-type estimate

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

Published

2014-03-25

Issue

Vol. 89 No. 3 (2014)

Section

Articles
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