Nikodým maximal function with restricted directions

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 $α>0$.