Nikodým maximal function with restricted directions

TL;DR

This paper analyzes the planar Nikodým maximal operator with direction restrictions $\mathcal N_{\Theta;δ}$ by connecting its $L^p$-boundedness to the fractal geometry of the direction set via the quasi-Assouad dimension $\dim_{\mathrm{qA}}\Theta = s$. Employing a sharp incidence bound for dyadic tubes (built from a 2-ends Furstenberg-set framework) and a careful discretization, the authors prove that when $s\in[\tfrac{1}{2},1]$, the critical exponent is $p_{\Theta}=1+s$, with $\mathcal N_{\Theta;δ}$ bounded on $L^p$ up to a $δ^{-ε}$ loss for $p\ge 1+s$ and unbounded for $p<1+s$; for $s<\tfrac{1}{2}$ the bound may fail (and they construct a counterexample with $s=1/3$). As an application, they exhibit a convex domain with affine dimension $\kappa_{\Omega}=\tfrac{1}{6}$ for which the Bochner–Riesz means converge in $L^6$ for all $α>0$, and they obtain refined, ε-dependent BR bounds (Theorems A and B). The work advances understanding of directional maximal operators, incidence geometry, fractal sets, and their consequences for Fourier multipliers. Overall, the results rigorously tie geometric/dimensional properties of direction sets to sharp $L^p$-bounds and BR convergence phenomena in the plane.

Abstract

We study the planar Nikodým maximal operator $\mathcal{N}_{Θ;δ}$ associated to a direction set $Θ\subset \mathbb{S}^{1}$. We show that the quasi-Assouad dimension $s := \dim_{\mathrm{qA}} Θ$ characterises the essential $L^{p}$-boundedness of $\mathcal{N}_{Θ;δ}$ in the following sense. If $s \in [\tfrac{1}{2},1]$, then $\mathcal{N}_{Θ;δ}$ is essentially bounded on $L^{p}(\mathbb{R}^{2})$ for $p \geq 1 + s$, and essentially unbounded for $p < 1 + s$. Here essential boundedness means $L^{p}$-boundedness with constant $O_ε(δ^{-ε})$. We also show that the characterisation described above fails for $s < \tfrac{1}{2}$. More precisely, there exists a set $Θ\subset \mathbb{S}^{1}$ with $\dim_{\mathrm{qA}} Θ= \tfrac{1}{3}$ such that $\mathcal{N}_{Θ;δ}$ is essentially unbounded on $L^{p}(\mathbb{R}^{2})$ for all $p < \tfrac{3}{2}$. As an application, we show there exists a convex domain with affine dimension $\tfrac{1}{6}$ such that the $α$-order Bochner-Riesz means converge in $L^6$ for all $α>0$.

Figures (2)

• Figure 1: A comparison of Theorem \ref{['theorem: seeger--ziesler']} and Theorem \ref{['theorem: A']}. Theorem \ref{['theorem: seeger--ziesler']} says that $B^\alpha_\Omega$ is bounded on $L^p$ for all $(1/p,\alpha)\in A$ for any $\Omega$ with $\kappa_\Omega=1/6$. Theorem \ref{['theorem: A']} constructs a specific $\Omega$ with $\kappa_\Omega=1/6$ such that $B^\alpha_\Omega$ is bounded on $L^p$ for all $(1/p,\alpha)\in A\cup B$.
• Figure 2: On the left: the set $P$ and the tubes $\mathcal{T}$. On the right: the dual squares of the tubes $\mathcal{T}$.

Theorems & Definitions (95)

• Theorem 1.1: Córdoba Cordoba_Nikodym
• Remark 1.2
• Theorem 1.3
• Corollary 1.4
• Remark 1.5
• Theorem 1.6: Córdoba Cordoba_Nikodym
• Theorem 1.7
• Remark 1.8
• Definition 1.11: Katz--Tao $(\delta,t,C)$-set
• Definition 1.12: $(\delta,s,C)$-regular set