STRONG TYPE INEQUALITIES AND AN ALMOST-ORTHOGONALITY PRINCIPLE FOR FAMILIES OF MAXIMAL OPERATORS ALONG DIRECTIONS IN R^2

The paper proves an almost-orthogonality principle for maximal operators over arbitrary sets of directions in R^2 . Namely, the L^p -bounds for an operator of this type are obtained from the corresponding L^p -bounds of the maximal functions associated to a certain partition of the set of directions, and from the particular structure of this partition. Applications to several types of maximal operators are provided.