Almost-orthogonality principles for certain directional maximal functions

TL;DR

This work delivers sharp $L^2$-bounds for directional maximal operators $M_\Omega$ and directional singular integral maximal operators $T_\Omega$ in $\mathbb{R}^n$ when the direction set $\Omega$ is equidistributed on the sphere. It introduces almost-orthogonality principles that quantify cross-direction contributions via a cap-based frequency decomposition and scale-dependent interaction counts $E_l$, enabling a divide-and-conquer inductive argument across scales. The main results show $||M_\Omega||_{L^2} \lesssim (#\Omega)^{\frac{n-2}{2(n-1)}}$ and $||T_\Omega||_{L^2} \lesssim (#\Omega)^{\frac{n-2}{2(n-1)}}$ for $n\ge 3$, with sharpness in general. The paper further extends the framework to mixed lacunary/equidistributed direction sets and generalizes prior 2D results to higher dimensions, offering a robust toolkit for directional maximal operators.

Abstract

We develop almost-orthogonality principles for maximal functions associated with averages over line segments and directional singular integrals. Using them, we obtain sharp $L^2$-bounds for these maximal functions when the underlying direction set is equidistributed in $\mathbb{S}^{n-1}$.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Lemma 3.1: cf. DM
• proof
• Lemma 3.2: cf. DM
• proof
• Lemma 3.3: cf. Alf
• proof
• proof : Proof of Theorem \ref{['thm:ortho']}