bounds for certain Bochner-Riesz multipliers.

Very recently, Culiuc, di Plinio and Ou [@160305317] have established a sparse domination result in a setting far removed from the extensions above: The trilinear form associated to the bilinear Hilbert transform is dominated by a sparse form. This is a surprising result, as the bilinear Hilbert transform has all the difficult features of the Hilbert transform, with additional oscillatory and arithmetic-like aspects. This paper is an initial effort on our part to understand how general a technique ‘domination by sparse’ could be. There are plenty of additional directions that one could think about.

For instance, the interest in the oscillatory singular integrals is driven in part by their application to singular integrals defined on nilpotent groups. Implications of the sparse bound in this setting are unexplored.

There are two approaches to sparse bounds, the bilinear form method [@160305317], and the use of the maximal truncation inequality [@150105818]. We use neither approach. After applying the known sparse bounds for singular integrals, for the remaining parts of the operator, there is a very simple interpolation argument which you can use in the bilinear setting. The notable point about the proofs are that they are quite easy, 