We prove new ℓ p (Z d ) bounds for discrete spherical averages in dimensions d ≥ 5. We focus on the case of lacunary radii, first for general lacunary radii, and then for certain kinds of highly composite choices of radii. In particular, if A λ f is the spherical average of f over the discrete sphere of radius λ, we havefor any lacunary sets of integers {λ 2 k }. We follow a style of argument from our prior paper, addressing the full supremum. The relevant maximal operator is decomposed into several parts; each part requires only one endpoint estimate.

The Bochner-Riesz multipliers B δ on R n are shown to satisfy a range of sparse bounds, for all 0 < δ < n−1 2 . The range of sparse bounds increases to the optimal range, as δ increases to the critical value, δ = n−1 2 , even assuming only partial information on the Bochner-Riesz conjecture in dimensions n ≥ 3. In dimension n = 2, we prove a sharp range of sparse bounds. The method of proof is based upon a 'single scale' analysis, and yields the sharpest known weighted estimates for the Bochner-Riesz multipliers in the category of Muckenhoupt weights.

Consider the discrete quadratic phase Hilbert Transform acting on ℓ 2 finitely supported functionsWe prove that, uniformly in α ∈ T, there is a sparse bound for the bilinear form H α f, g . The sparse bound implies several mapping properties such as weighted inequalities in an intersection of Muckenhoupt and reverse Hölder classes.

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In this paper we prove that the space of two parameter, matrix-valued BMO functions can be characterized by considering iterated commutators with the Hilbert transform. Specifically, we prove thatThe upper estimate relies on Petermichl's representation of the Hilbert transform as an average of dyadic shifts, and the boundedness of certain paraproduct operators, while the lower bound follows Ferguson and Lacey's proof for the scalar case.