Bilinear Spherical Maximal Functions of Product Type

Abstract

In this paper we introduce and study a bilinear spherical maximal function of product type in the spirit of bilinear Calderón-Zygmund theory. This operator is different from the bilinear spherical maximal function considered by Geba et al. in [14]. We deal with lacunary and full versions of this operator, and prove weighted estimates with respect to genuine bilinear weights beyond the Banach range. Our results are implied by sharp sparse domination for both the operators, following ideas by Lacey [20]. In the case of the lacunary maximal operator we also use interpolation of analytic families of operators to address the weighted boundedness for the whole range of tuples.

Sparse bounds for the bilinear spherical maximal function

“…It is also of interest to study the lacunary version of the spherical maximal function, both in the linear and bilinear setting. Indeed, there is an analogous result obtained for the lacunary version of the operator in both [Lac19] and [RSS21]. In our case, the natural object is the following lacunary bilinear spherical maximal function…”

“…First, our operator doesn't have a product structure, making it difficult to deduce multilinear estimates from their linear counterparts. This is in contrast to the situation in [RSS21], where the product type bilinear spherical maximal function…”

“…However, it seems that our method doesn't imply an improved result (i.e. larger sparse range) for M lac , unlike the situations in [Lac19,RSS21]. This is because our proof involves slicing the bilinear operator, which would result in a linear single scale averaging operator for a continuum spectrum of radii.…”

“…Indeed, even though there is already quite some success in obtaining sparse bounds for multilinear operators such as Calderón-Zygmund operators [CAR16,LN19], rough singular integrals [Bar17], and bilinear Hilbert transforms [CDPO18,BM21], the investigation for multilinear Radon transforms only started very recently, before which the weighted norm inequalities for such operators (with respect to multilinear Muckenhoupt weights) were far out of reach. To date, the only known sparse domination results for such operators are due to [RSS21] and [PS21], where the product type bilinear spherical maximal function and the triangle averaging operator are studied, respectively. The multilinear operators under study here are natural analogues of important linear Radon type operators.…”

“…It is shown in [JL20] that for any tuple (p, q, r) such that ( 1 p , 1 q , 1 r ) ∈ R(d) (or part of the boundary of the region), M maps from L p × L q into L r , which is often referred to as the L p improving estimate, as the target space L r is better than the space corresponding to the Hölder exponent. Following closely the methods in [RSS21,PS21], one can easily reduce the desired sparse bounds to the continuity L p improving estimates of the form…”