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Pelczyriski, Absolutely summing operators in 13, spaces and applications. Studia Mathematica, 29 (1968), 275-326. 16. L. Schwartz, Un mathimaticien aux prises awec le sikcle. Editions Odile Jacob, 1997. 37 THE HARDY-LITTLEWOOD MAXIMAL FUNCTION AND SOME OF ITS VARIANTS JAVIER DUOANDIKOETXEA* Departamento de Matema'ticas, Universidad del Pais Vasco-Euskal Herriko Unibertsitatea, Apartado 644, 48080 Bilbao (Spain) E-mail: javier. duoandikoetxea@ehu. es The Hardy-Littlewood maximal function introduced in 1930 has several applications, the most popular being Lebesgue's differentiation theorem.

L ) f ( Y )dY (kernel operator) had especially good properties, but unfortunately they did not exhaust all the possible operators. And by the way, this was one of the main difficulties in the rigorous formulation of Quantum Mechanics. The idea of P. Dirac and others was to express any observable (N ‘(linear operator”) in terms of a base formed by the “states” (11,)of the system (the eigenfunctions corresponding to the eigenvalues of the operator, that is, the “spectrum” of the observable). But in most cases this spectrum was not countable (in words of Dirac: “...

The generalization of the differentiation theorem to averages over a variety of families of sets leads to the definition of several variants of the Hardy-Littlewood maximal operator. Important cases are averages over balls or cubes (the usual Hardy-Littlewood maximal function) , averages over rectangles with sides parallel to the coordinate axes, and averages over arbitrary rectangles. A review of these operators and of the concept of differentiation basis will be the subject of the first part of this paper.