Sparse domination of Hilbert transforms along curves

Abstract

We obtain sharp sparse bounds for Hilbert transforms along curves in R n , and derive as corollaries weighted norm inequalities for such operators. The curves that we consider include monomial curves and arbitrary C n curves with nonvanishing torsion. Definition 1.1. A collection of γ-cubes S ⊂ Q γ in R n is δ-sparse for 0 < δ < 1 if there exist sets {E S : S ∈ S} that are pairwise disjoint, E S ⊂ S, and satisfy |E S | > δ|S| for all S ∈ S.2010 Mathematics Subject Classification. Primary: 42B20, Secondary: 42B99.

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;…”

“…This lemma is sharp at the level of sparse bounds and provides the best possible dependence on the A p characteristic of the weight. The intersection of the A p and the reverse Hölder classes may be understood as (12) w P A p{r X RH ps{pq 1 ðñ w ps{pq 1 P A ps{pq 1 pp{r´1q`1 .…”

Sparse domination of singular Radon transform

“…The second part is to find an approatiate approach to apply the sparse domination principle to the dyadic systems constructed in the previous step. Main difficulities under the current situation are that this space of homogeneous type lacks both a dilation structure and a group strucure, in particular, we are not able 5 For general space of homogeneous type, the associated measure can be any non-negative doubling Borel measure. In this paper, it suffices for us to consider the case when such a measure is Lebesuge.…”

“…and d(I) = d i1 + · · · + d in and Vol(A) denotes the induced Lebesgue volume on the leaf generated by the Y j s, passing through the point x 0 ; (5). As a consequence of (4), we have for every compact set Ω ⋐ Ω ′ , there is a constant C Ω so that if x ∈ Ω and if δ < δ0 2 , Finally, it will be convenient to assume that the ball B (Y,d) (x, δ) lies "inside" of Ω in the following sense.…”

“…This began with the work of Lacey [8] on spherical maximal functions and then was furthered by Oberlin [14] by considering convolution operators induced by general compactly supported measures with Fourier decay. The first attempt where the underlying dilation structure is nonisotroptic was due to Cladek and Ou [5], where they studied the sparse bound of the Hilbert transform along the monomial curve. More precisely, given γ : R → R n a monomial curve, that is γ(t) := (|t| α1 , .…”