Inequalities for some maximal functions. I

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.