Improved bounds for the bilinear spherical maximal operators

On families between the Hardy–Littlewood and spherical maximal functions

“…We pick sequences ϕ k j of smooth compactly supported functions with ϕ k j →g j in L qj (R n ) (since q j <∞) and consider the sequence 15) if α<1 (or by [8] if α=1) this sequence converges to zero in L q (R n ), thus there is a subsequence that converges to zero a.e. This implies that there is a subset E of R n of measure zero such that for all x∈R n \E we have…”

“…When m=1, S m reduces to S in (1). The bilinear analogue of Stein's spherical maximal function (when m = 2) was first introduced in [10] by Geba, Greenleaf, Iosevich, Palsson, and Sawyer who obtained the first bounds for it but later improved bounds were provided by [3], [12], [15] and [17]. A multilinear (non-maximal) version of this operator when all input functions lie in the same space L p (R) was previously studied by Oberlin [21].…”

“…Let n≥2, 0≤α<1, and 1<p i ≤∞. Define p by m i=1 Moreover, if (15) holds, then the constant C can be chosen to be independent of the dimension (as indicated by the parameters on which it is claimed to depend).…”

Discrete multilinear maximal functions and number theory

“…A prototypical curved object is the sphere, and maximal spherical averaging operators arise naturally in many contexts. From the multilinear view, optimal Lebesgue space bounds for (multilinear) spherical maximal functions had been pursued in many papers, such as [6,10,11,22], building upon work of Stein [23] and Bourgain [7]. From a discrete perspective, Magyar-Stein-Wainger showed optimal bounds that were both different from the continuous ones and heavily employed number theoretic techniques [19].…”

Maximal estimates for the bilinear spherical averages and the bilinear Bochner-Riesz operators

“…The operator M first appeared in [19] and, subsequently, studied by Barrionevo-Grafakos-D.He 1 -Honzík-Oliveira [3], Grafakos-D.He-Honzík [20], and Heo-Hong-Yang [24]. Let 1 ≤ p, q ≤ ∞ and 0 < r ≤ ∞ satisfy the Hölder relation…”