A note on maximal averages in the plane (original) (raw)

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In this paper we give a simple proof of a long-standing conjecture, recently proved by N. Katz, on the weak-type norm of a maximal operator associated with an arbitrary collection of directions in the plane. The proof relies upon a geometric argument and on induction with respect to the number of directions. Applications are given to estimate the behavior of several types of maximal operators.

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In this paper we give a simple proof of a long-standing conjecture, recently proved by N. Katz, on the weak-type norm of a maximal operator associated with an arbitrary collection of directions in the plane. The proof relies upon a geometric argument and on induction with respect to the number of directions. Applications are given to estimate the behavior of several types of maximal operators.

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SUNTO. Si presentono alcuni risultati sugli operatori massimali associati alle misure superficiali in R n . E. M. Stein [9] ha iniziato lo studio di quest'argomento, dimostrando una disuguaglianza a priori per la funzione massimale sferica, mediante l'utilizzo delle funzioni "g". In [3], M. Cowling e G. Mauceri hanno generalizzato il risultato di Stein. Qui si dà una dimostrazione diversa e forse più semplice della stima a priori, utilizzando la transformata di Mellin, come suggerito in un altro lavoro di Cowling e Mauceri [2].

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