Nikodým maximal function with restricted directions

Abstract

We study the planar Nikodým maximal operator ${\mathcal{N}}_{\hat{I}˜;\hat{I}´}$ associated to a direction set $\hat{I}˜\subset {\mathbb{S}}^1$. We show that the quasi-Assouad dimension $s:={\dim }_{\mathrm{qA}}\hat{I}˜$ characterises the essential $L^p$-boundedness of ${\mathcal{N}}_{\hat{I}˜;\hat{I}´}$ in the following sense. If $s\in [\frac{1}{2},1]$, then ${\mathcal{N}}_{\hat{I}˜;\hat{I}´}$ 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 $O_{\hat{I}}\mu (\hat{I}{´}^{-\hat{I}\mu })$. We also show that the characterisation described above fails for $s<\frac{1}{2}$. More precisely, there exists a set $\hat{I}˜\subset {\mathbb{S}}^1$ with ${\dim }_{\mathrm{qA}}\hat{I}˜=\frac{1}{3}$ such that ${\mathcal{N}}_{\hat{I}˜;\hat{I}´}$ 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 $\hat{I}\pm$-order Bochner-Riesz means converge in $L^6$ for all $\hat{I}\pm >0$.

28 pages, 2 figures