Acta Acad. Paed. Agriensis Sectio Matematicae 30 (2003), 55-66. ON THE L 1 NORM OF THE WEIGHTED MAXIMAL FUNCTION OF THE WALSH-KACZMARZ-DIRICHLET KERNELS GYÖRGY GÁT This paper is dedicated to the memory of Professor Péter Kiss Abstract. In this paper we investigate the integral of the weighted maximal function of the Walsh-Paley-Dirichlet, and the Walsh-Kaczmarz-Dirichlet kernels. We find necessary and sufficient conditions for the finiteness of the integrals. The conditions are quite different for the two rearrangements of the Walsh system. The Walsh system in the Kaczmarz enumeration was studied by a lot of authors (see [4], [5], [8], [7], [1], [6], [9]). In [2] it has been pointed out that the behavior of the Dirichlet kernel of the Walsh-Kaczmarz system is worse than of the kernel of the Walsh-Paley system considered more often. Namely, it is proved [2] that for the Dirichlet kernel D n (x) of the Walsh- Kaczmarz system the inequality lim sup n→∞ |D n (x)| log n ≥ C> 0 holds a.e. This “spreadness” of this system makes easier to construct examples of divergent Fourier series [1]. A number of pathological properties is due to this “spreadness” property of the kernel. For example, for Fourier series with respect to the Walsh-Kaczmarz system it is impossible to establish any local test for convergence at a point or on an interval, since the principle of localization does not hold for this system. On the other hand, the global behavior of the Fourier series with respect to this system is similar in many aspects to the case of the Walsh-Paley system. Schipp [5] and Wo-Sang Young [9] proved that the Walsh-Kaczmarz system is a convergence system. Skvorcov [8] verified the everywhere (and uniform) convergence of the Fejér means of continuous functions, and Gát proved [3] that the Fejér-Lebesgue theorem also holds for the Walsh-Kaczmarz system. Beyond the convergence theorems of the Fourier series one can often find some boundedness properties of the Dirichlet kernel functions. For instance, for the Walsh-Paley system we have sup n∈N |D n (x)| < ∞ for each x 6=0. This - as we have seen above - is not the case for the Kaczmarz rearrangement. What can be said for the norm of maximal functions? It is easy to have that the L 1 norm of sup n∈N |D n | with respect to both systems is infinite. What happens if we apply some weight function α? That is, on what conditions find we the inequality     sup n∈N fl fl fl fl D n α(n) fl fl fl fl     1 < ∞ Date : Jan. 15, 2003. 1991 Mathematics Subject Classification. 42C10. Key words and phrases. Walsh-Paley, Walsh-Kaczmarz system, Dirichlet kernels, weighted maximal func- tions, integral. Research supported by the Hungarian “Művelődési és Közoktatási Minisztérium”, grant no. FKFP 0182/2000, by the Hungarian National Foundation for Scientific Research (OTKA), grant no. M 36511/2001. 1