Nikodym sets and maximal functions associated with spheres

TL;DR

This work develops a spherical analogue of Nikodym sets and analyzes associated maximal operators. It establishes sharp $L^p$ bounds for $\mathcal{N}^\delta$ and $\mathcal{S}^\delta$ via detailed geometric intersection estimates of $\delta$-neighborhoods of spheres, with consequences that Nikodym sets for spheres have full Hausdorff dimension. The paper further analyzes uncentered maximal operators $\mathcal{N}_T$ and $\mathcal{S}_T$, showing boundedness results when $T$ has finite upper Minkowski content below $n-1$ and connecting fractal bounds to fractal local smoothing estimates for the wave equation. It then extends to a broad class of spherical maximal operators $M^\delta_{\mathcal{C}}$ and provides a Littlewood–Paley framework to study $N_T$ and $S_T$, including fractal spherical averages with respect to measures in $\mathcal{C}(\alpha)$. Finally, the authors present sharp lower bounds demonstrating the tightness of their $L^p$-bounds and clarify the dependence on dimension, fractal parameters, and the translating set, complementing the upper-bound theory with geometric constructions inspired by Kakeya/Nikodym phenomena.

Abstract

We study spherical analogues of Nikodym sets and related maximal functions. In particular, we prove sharp $L^p$-estimates for Nikodym maximal functions associated with spheres. As a corollary, any Nikodym set for spheres must have full Hausdorff dimension. In addition, we consider a class of maximal functions which contains the spherical maximal function as a special case. We show that $L^p$-estimates for these maximal functions can be deduced from local smoothing estimates for the wave equation relative to fractal measures.

Figures (7)

• Figure 1: Range of boundedness of $\mathcal{N}_{T}$ in the case $n=5$. The blue and red lines indicate the boundaries of sufficient and necessary conditions, respectively.
• Figure 2: Range of boundedness of $\mathcal{S}_{T}$ in the case $n=5$. The blue and red lines indicate the boundaries of sufficient and necessary conditions, respectively.
• Figure 3: Table of examples for $\mathcal{N}^\delta$. The sets $E$ and $Z$ satisfy \ref{['eq:EZ-good']}. The quantities $\alpha, \beta, \gamma$ are defined as in \ref{['eq:alpha-beta-gamma-def']}. (See \ref{['figure:cyl-shell-rad-1/sqrt2']} for some pictures.)
• Figure 4: The last two examples from \ref{['figure:examples-NM']}.
• Figure 5: Table of examples for $\mathcal{S}^\delta$. The sets $E$ and $Z$ satisfy \ref{['eq:EZ-good2']}. The quantities $\alpha, \beta, \gamma$ are defined as in \ref{['eq:alpha-beta-gamma-def2']}.

Theorems & Definitions (87)

• Theorem 1.1: Existence of Nikodym sets for unit spheres
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Proposition 1.8
• Remark 1.9
• Theorem 1.10