Singular integrals along lacunary directions in ${\mathbb{R}}^n$

Abstract

A recent result by Parcet and Rogers is that finite order lacunarity characterizes the boundedness of the maximal averaging operator associated to an infinite set of directions in ${\mathbb{R}}^n$. Their proof is based on geometric-combinatorial coverings of fat hyperplanes by two-dimensional wedges. Seminal results by Nagel-Stein-Wainger relied on geometric coverings of n-dimensional nature. In this article we find the sharp cardinality estimate for singular integrals along finite subsets of finite order lacunary sets in all dimensions. Previous results only covered the special case of the directional Hilbert transform in dimensions two and three. The proof is new in all dimensions and relies, among other ideas, on a precise covering of the n-dimensional Nagel-Stein-Wainger cone by two-dimensional Parcet-Rogers wedges.

arXiv comment18 pages, 1 figure. Final version to appear in Adv. Math

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• $L^2$ Bounds for a maximal directional Hilbert transform