A Stein Conjecture for the Circle (original) (raw)

The circular maximal operator on Heisenberg radial functions

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 2021

Lebesgue space estimates are obtained for the circular maximal function on the Heisenberg group H 1 restricted to a class of Heisenberg radial functions. Under this assumption, the problem reduces to studying a maximal operator on the Euclidean plane. This operator has a number of interesting features: it is associated to a non-smooth curve distribution and, furthermore, fails both the usual rotational curvature and cinematic curvature conditions.

On weighted estimates for Stein's maximal function

Bulletin of The Australian Mathematical Society, 1998

In this brief article, we prove some weighted estimates for Stein's maximal function by using interpolation techniques. In some cases, our results agree with those previously obtained in and .

On maximal functions over circular sectors with rotation invariant measures

Commentationes Mathematicae Universitatis Carolinae

Given a rotation invariant measure in R n , we define the maximal operator over circular sectors. We prove that it is of strong type (p, p) for p > 1 and we give necessary and sufficient conditions on the measure for the weak type (1, 1) inequality. Actually we work in a more general setting containing the above and other situations.

A sharp trilinear inequality related to Fourier restriction on the circle

Revista Matemática Iberoamericana

In this paper we prove a sharp trilinear inequality which is motivated by a program to obtain the sharp form of the L 2 − L 6 Tomas-Stein adjoint restriction inequality on the circle. Our method uses intricate estimates for integrals of sixfold products of Bessel functions developed in a companion paper [24]. We also establish that constants are local extremizers of the Tomas-Stein adjoint restriction inequality as well as of another inequality appearing in the program.

A Note on a Maximal Function Over Arbitrary Sets of Directions

Bulletin of the London Mathematical Society, 2000

We consider a sequence of operators over sets of finite equidistributed directions converging to S. We provide a new proof of N. Katz's bound for such operators. As a corollary, we deduce that S is bounded from some subsets of L# to L#. These subsets are composed of positive functions whose Fourier transforms have a logarithmic decay or which are supported on a disc.

Inequalities for Some Maximal Functions. I

Transactions of The American Mathematical Society, 1985

This paper presents a new approach to maximal functions on R". 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 L2(R2). 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 > 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, 4] to which N. E. Aguilera and E. O. Harboure [2] had provided a partial response.

Maximal Functions and Measures on the Upper-Half Plane

Analysis Mathematica, 2018

We study weighted boundedness of Hardy-Littlewood-type maximal function involving Orlicz functions. We also obtain some sufficient conditions for the weighted boundedness of the Hardy-Littlewood maximal function of the upper-half plane. 1 |Q I |ω,α Q I |f (z)| p ω(z)dV α (z)