Nikodým maximal function with restricted directions

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