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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Inequalities for some maximal functions. I

Authors: Michael Cowling and Giancarlo Mauceri
Journal: Trans. Amer. Math. Soc. 287 (1985), 431-455
MSC: Primary 42B25; Secondary 42B20
MathSciNet review: 768718
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Abstract: This paper presents a new approach to maximal functions on ${{\mathbf {R}}^n}$. Our method is based on Fourier analysis, but is slightly sharper than the techniques based on square functions. In this paper, we reprove a theorem of E. M. Stein [16] on spherical maximal functions and improve marginally work of N. E. Aguilera [1] on the spherical maximal function in ${L^2}({{\mathbf {R}}^2})$. We prove results on the maximal function relative to rectangles of arbitrary direction and fixed eccentricity; as far as we know, these have not appeared in print for the case where $n \geqslant 3$, though they were certainly known to the experts. Finally, we obtain a best possible theorem on the pointwise convergence of singular integrals, answering a question of A. P. Calderón and A. Zygmund [3,3] to which N. E. Aguilera and E. O. Harboure [2] had provided a partial response.

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Keywords: Maximal functions, Mellin transform, spherical harmonics
Article copyright: © Copyright 1985 American Mathematical Society