Bounds for discrete multilinear spherical maximal functions in higher dimensions

Abstract

We find the sharp range for boundedness of the discrete bilinear spherical maximal function for dimensions d⩾5. That is, we show that this operator is bounded on lpfalse(Zdfalse)×lqfalse(Zdfalse)→lrfalse(Zdfalse) for 1/p+1/q⩾1/r and r>d/(d−2) and we show this range is sharp. Our approach mirrors that used by Jeong and Lee in the continuous setting. For dimensions d=3,4, our previous work, which used different techniques, still gives the best known bounds. We also prove analogous results for higher degree k, ℓ‐linear operators.

Bilinear Spherical Maximal Functions of Product Type

“…For notational convenience we shall restrict ourselves to the bilinear setting in this paper. Given locally integrable functions f 1 and f 2 defined on R n , the bilinear maximal function M(f 1 , f 2 ) is defined by (1) M(f 1 , f 2 )(x) := sup…”

“…In this paper we introduce a bilinear analogue of the spherical maximal function in the spirit of the bilinear Hardy-Littlewood maximal function (1), which plays a key role in the theory of bilinear Calderón-Zygmund operators. Define…”

“…This result is stated in Theorem 2.1. We exploit the ideas from [20] and establish a sparse domination principle for the bilinear spherical maximal functions in Theorem 2.3 so 1 In a private communication with the second and third authors, Loukas Grafakos suggested that the terminology for this operator should be instead maximal product of spherical averages to better portray the nature of the operators.…”

“…and extended the L p 1 × L p 2 → L p estimates for the operator M sph to the best possible range of exponents p 1 , p 2 and p for all n ≥ 2 (note that an estimate similar to (3) holds with the roles of M full and M interchanged due to symmetry). We also refer to the recent papers [1,11] for the generalisation of the bilinear spherical maximal function to the multilinear setting. Weighted estimates for the bilinear maximal operator M sph defined in (2) beyond the ones that can be obtained trivially from the pointwise estimate (3) remain as an open problem.…”

Lp estimates for multilinear convolution operators defined with spherical measure

“…We shall fill this gap in this paper. We also refer to the recent papers [1,6,14] for further generalization of the bilinear spherical maximal functions to the multilinear and product type setting.…”