Nikodým maximal function with restricted directions

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