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

Abstract

We study the maximal estimates for the bilinear spherical average and the bilinear Bochner-Riesz operator. First, we obtain L p × L q → L r estimates for the bilinear spherical maximal function on the optimal range. Thus, we settle the problem which was previously considered by Geba, Greenleaf, Iosevich, Palsson and Sawyer, later Barrionevo, Grafakos, D. He, Honzík and Oliveira, and recently Heo, Hong and Yang. Secondly, we consider L p × L q → L r estimates for the maximal bilinear Bochner-Riesz operators and improve the previous known results. For the purpose we draw a connection between the maximal estimates and the square function estimates for the classical Bochner-Riesz operators.

Multilinear Spherical Maximal Function

“…As was noted in [18], the method used in the proof of Theorem 1 also yields bounds for the stronger multi(sub)linear operator…”

“…In this work we extend the results of Jeong and Lee in the multilinear setting and we adapt the counterexample of Barrionuevo, Grafakos, He, Honzík, and Oliveira in [4] to show that our results are sharp. We also provide a counterexample that addresses a question raised by Jeong and Lee in [18] regarding the validity of a strong type L 1 × L ∞ → L 1 bound for the bilinear spherical maximal function.…”

“…The bi(sub)linear analogue of Stein's spherical maximal function was first introduced in by Geba, Greenleaf, Iosevich, Palsson, and Sawyer [11] who obtained the first bounds for it but later improved bounds were provided by [4], [14], [17], and [18]. 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].…”

“…The authors in [4] provided an example that shows that the bilinear spherical maximal function is not bounded when p ≥ n 2n−1 . Earlier this year, Jeong and Lee in [18] proved that the bilinear maximal function is pointwise bounded by the product of the linear spherical maximal function and the Hardy-Littlewood maximal function, which helped them establish boundedness in the optimal open set of exponents, along with some endpoint estimates. Recently certain analogous bounds have been obtained by Anderson and Palsson [1] [2] concerning the discrete multilinear spherical maximal function.…”

“…The following proposition provides a negative answer to a question posed by Jeong and Lee in [18] regarding a strong type…”