A REMARK ON MAXIMAL OPERATORS ALONG DIRECTIONS IN
ANGELES ALFONSECA, FERNANDO SORIA, AND 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.
Let be a subset of [0,
4 ). Associated to it, we define the maximal operator M
acting on locally integrable functions f on R2 by
Mf(x) = sup
where B denotes the basis of all rectangles with longest side forming an angle with
the x-axis, for some .