Mathematical Research Letters

Volume 10 (2003)

Number 1

A remark on maximal operators along directions in ${\Bbb R}^2$

Pages: 41 – 49

DOI: http://dx.doi.org/10.4310/MRL.2003.v10.n1.a5

Authors

Angeles Alfonseca

Fernando Soria

Ana Vargas

Abstract

In this paper we give a simple proof of a long-standing conjecture, recently proved by N. Katz, on the weak-type norm of a maximal operator associated with an arbitrary collection of directions in the plane. The proof relies upon a geometric argument and on induction with respect to the number of directions. Applications are given to estimate the behavior of several types of maximal operators.

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