On the maximal directional Hilbert transform in three dimensions

Abstract

We establish the sharp growth rate, in terms of cardinality, of the $L^p$ norms of the maximal Hilbert transform $H_{\Omega }$ along finite subsets of a finite order lacunary set of directions $\Omega \subset {\mathbb{R}}^3$, answering a question of Parcet and Rogers in dimension $n=3$. Our result is the first sharp estimate for maximal directional singular integrals in dimensions greater than 2. The proof relies on a representation of the maximal directional Hilbert transform in terms of a model maximal operator associated to compositions of two-dimensional angular multipliers, as well as on the usage of weighted norm inequalities, and their extrapolation, in the directional setting.

26 pages, 2 figures, final version, to appear in Int. Math. Res. Not. IMRN

References in corpus (3)

• Square functions for bi-Lipschitz maps and directional operators
• A sharp estimate for the Hilbert transform along finite order lacunary sets of directions
• On the maximal directional Hilbert transform

Cited by in corpus (5)

• Maximal directional operators along algebraic varieties
• Paraproducts for bilinear multipliers associated with convex sets
• $L^2$ Bounds for a maximal directional Hilbert transform
• Singular integrals along lacunary directions in ${\mathbb{R}}^n$
• Almost-orthogonality principles for certain directional maximal functions