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**Additional info for Advanced Courses of Mathematical Analysis II: Proceedings of the Second International School, Granada, Spain, 20 - 24 September 2004**

**Sample text**

H e s z (1932) in which he used his “rising sun lemma” to obtain the weak inequality. We refer to [44] or [2] for the use of rearrangements; the proof of F. Riesz appears in 12’], [16] and [44], for instance. The rising sun lemma gives for any of the one-sided versions of the maximal operator (3) the following equality: i r A different type of proof by using a covering lemma will be better adapted to higher dimensions. The convenient form in the one-dimensional case is the following. 1. Given a finite family of intervals of the real line, there is a subfamily with the same union and such that each point is at most in two intervals of the subfamily.

The total number of independent states is infinite, and equal t o the number of points of a line” ). Then the corresponding “matrix” ( a p q was ) indexed by two continuous parameters, p , q E R,and the action of the observable cy on some state qq was written in the form that is, as a kernel operator. But then, if one wants to represent in this way the “multiplication by a non zero constant c” operator, we need a kernel of the type cypq = c6(p - q ) , that is, a singular function. Dirac used also other 25 singular functions and its derivatives, just applying formally the method of “integration by parts”.

15. J. Lindenstrauss and A. 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.