Sparse domination of singular Radon transform

Abstract

The purpose of this paper is to study the sparse bound of the operator of the form f → ψ(x) f (γt(x))K(t)dt, where γt(x) is a C ∞ function defined on a neighborhood of the origin in (x, t) ∈ R n × R k , satisfying γ 0 (x) ≡ x, ψ is a C ∞ cut-off function supported on a small neighborhood of 0 ∈ R n and K is a Calderón-Zygmund kernel suppported on a small neighborhood of 0 ∈ R k . Christ, Nagel, Stein and Wainger gave conditions on γ under which T : L p → L p (1 < p < ∞) is bounded. Under the these same conditions, we prove sparse bounds for T , which strengthens their result. As a corollary, we derive weighted norm estimates for such operators.

Sparse bounds for discrete singular Radon transforms

“…Sparse bounds for real-variable singular Radon transforms were studied in a much more general setting in [Hu20]. For our purpose, we need the following additional information on the sparse family: roughly speaking, if the support of K is far from the origin, then the cubes in the sparse collection should have large sidelength.…”

“…Here we use the concept of bilinear sparse bounds from [BFP16], [CDO18], [Lac19]. Recently, many new directions have been studied, including sparse domination of Radon transforms [CO18], [Obe19], [Hu20], discrete analogues [CKL19], [KL18], and operators on spaces of homogeneous type [AV14].…”

“…the associated (constant coefficient) vector fields span R n . This is needed for the application of sparse bounds for the real-variable operator from the second author's thesis [Hu20].…”

“…Another new difficulty when compared to [CKL19] is that the corresponding real-variable operator is no longer of Calderón-Zygmund type. Instead, we need to make use of the recently developed general theory for sparse domination of Radon transforms on R n (see [Hu20], and also [CO18] for an earlier result). To this end, we also remark that the pointwise sparse domination paradigm of [Ler13] does not apply.…”

“…In §3 we perform the decomposition of our operator into major and minor arc terms and explain how Theorem 1.2 is proven. In §4 we use Condition (Ł) to derive a variant of the real-variable sparse bound proven in [Hu20]. This is a key ingredient for the major arc estimate.…”