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Summary: AN ALMOST-ORTHOGONALITY PRINCIPLE IN L2 FOR
DIRECTIONAL MAXIMAL FUNCTIONS
ANGELES ALFONSECA, FERNANDO SORIA AND ANA VARGAS
Abstract. In this work we improve our result in [2]. We prove a
strong-type almost-orthogonality principle for maximal functions along
several directions. We use geometric methods and a covering lemma.
1. Introduction
Let be a subset of [0, ). Associated to we consider the basis B of
all rectangles in R2 whose longest side forms an angle with the x-axis, for
some . The maximal operator associated with the set is defined by
Mf(x) = sup
xRB
1
|R| R
|f(y)| dy.
The study of directional maximal functions began many years ago, and
some particular cases were studied by Str¨omberg [11], C´ordoba and Fef-
ferman [5], Nagel, Stein and Wainger [9], Sj¨ogren and Sj¨olin [10]. More
recently, the interest on these problems was renewed with the results of
Barrionuevo [3] and Katz [7, 8]. Nevertheless, only the operators associ-
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