Maximal subspace averages

TL;DR

This paper provides sharp $L^2$ bounds for maximal operators associated with averaging over subspaces in $\,\mathbb{R}^n$, extending known results from the line case to higher dimensions and codimensions.

Contribution

It systematically studies maximal subspace averages for all $1\leq d<n$, establishing sharp bounds and formulating a maximal Nikodym conjecture for higher dimensions.

Findings

Sharp $L^2$ bounds for maximal subspace averaging operators.

Critical weak $(2,2)$-bound in codimension 1 case.

Best possible $L^2$ bounds for the $(d,n)$-Nikodym maximal function.

Abstract

We study maximal operators associated to singular averages along finite subsets $\Sigma$ of the Grassmannian $\mathrm{Gr}(d,n)$ of $d$-dimensional subspaces of $\mathbb R^n$. The well studied $d=1$ case corresponds to the the directional maximal function with respect to arbitrary finite subsets of $\mathrm{Gr}(1,n)=\mathbb S^{n-1}$. We provide a systematic study of all cases $1\leq d<n$ and prove essentially sharp $L^2(\mathbb R^n)$ bounds for the maximal subspace averaging operator in terms of the cardinality of $\Sigma$, with no assumption on the structure of $\Sigma$. In the codimension $1$ case, that is $n=d+1$, we prove the precise critical weak $(2,2)$-bound. Drawing on the analogy between maximal subspace averages and $(d,n)$-Nikodym maximal averages, we also formulate the appropriate maximal Nikodym conjecture for general $1<d<n$ by providing examples that determine the critical $L^p$-space for the $(d,n)$-Nikodym problem. Unlike the $d=1$ case, the maximal Kakeya and Nikodym problems are shown not to be equivalent when $d>1$. In this context, we prove the best possible $L^2(\mathbb R^n)$-bound for the $(d,n)$-Nikodym maximal function for all combinations of dimension and codimension. Our estimates rely on Fourier analytic almost orthogonality principles, combined with polynomial partitioning, but we also use spatial analysis based on the precise calculation of intersections of $d$-dimensional plates in $\mathbb R^n$.