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

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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Published 1 January 2003