Sparse Bounds for Bochner–Riesz Multipliers

Abstract

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.

Sparse bounds for pseudodifferential operators

“…This approach has proved to be highly successful, as it applies to operators that fall well beyond the classical Calderón-Zygmund theory. Among many examples, we may find Bochner-Riesz multipliers [4,33], rough singular integrals [13], the bilinear Hilbert transform [16], the variational Carleson operator [19], oscillatory singular integrals [34,28], spherical maximal functions [30], a specific singular Radon transform [44] or the recent work by Ou and the second author [12] for Hilbert transforms along curves;…”

“…The methods to prove the above domination theorem are inspired by the recent works of Lacey and Spencer [34] and Lacey, Mena and Reguera [33]. It consists in obtaining geometrically decaying sparse bounds for a single scale version of the operators under study.…”

“…Indeed when a " 1, b " pn`1q{2`δ, the kernels K a,b are the convolution kernels associated to the Bochner-Riesz multipliers m δ . Sparse bounds for Bochner-Riesz multipliers have been recently obtained by Lacey, Mena and Reguera in [33]; see also the endpoint result of Kesler and Lacey [27] or the previous work of Benea, Bernicot and Luque [4]. Besides the weighted A p estimates obtained as a consequence of the sparse domination, one should note that there are weighted Fefferman-Stein inequalities for Bochner-Riesz multipliers involving Kakeya-type maximal functions; see for instance the earlier work of Carbery [8] or Carbery and Seeger [10].…”

Extrapolation of compactness on weighted Morrey spaces

“…In this section we will apply Theorem 1.8 to the commutators of Bochner-Riesz multipliers in dimensions d ≥ 2. Following [25,29], we recall that a Bochner-Riesz multiplier is a Fourier multiplier B κ with the symbol (1 − |ξ| 2 ) κ + , where κ > 0 and t + = max(t, 0). That is, the Bochner-Riesz operator is defined, on the class S(R d ) of Schwartz functions, by…”

Sparse domination for the lattice Hardy–Littlewood maximal operator

“…Since then, a wide variety of operators has been dominated by sparse operators (or, more generally, sparse forms). We refer the reader to the introductions, for example, in [1,4,25,21,26] for an overview of this vast field.…”