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Maximal subspace averages

Francesco Di Plinio, Ioannis Parissis

Abstract

We study maximal operators associated to singular averages along finite subsets $Σ$ 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 $Σ$, with no assumption on the structure of $Σ$. 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 $\mathbb R^n$.

Maximal subspace averages

Abstract

We study maximal operators associated to singular averages along finite subsets of the Grassmannian of -dimensional subspaces of . The well studied case corresponds to the the directional maximal function with respect to arbitrary finite subsets of . We provide a systematic study of all cases and prove essentially sharp bounds for the maximal subspace averaging operator in terms of the cardinality of , with no assumption on the structure of . In the codimension case, that is , we prove the precise critical weak -bound. Drawing on the analogy between maximal subspace averages and -Nikodym maximal averages, we also formulate the appropriate maximal Nikodym conjecture for general by providing examples that determine the critical -space for the -Nikodym problem. Unlike the case, the maximal Kakeya and Nikodym problems are shown not to be equivalent when . In this context, we prove the best possible -bound for the -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 -dimensional plates in .

Paper Structure

This paper contains 29 sections, 29 theorems, 216 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Thin $(d,n)$-averages for arbitrary finite $\Sigma\subset {\mathrm{Gr}(d,n)}$
  3. Sharp critical estimates for multiscale $(d,d+1)$-averages
  4. $L^2$-estimates for the $(d,n)$-Nikodym maximal operator
  5. Motivation and Background
  6. Methodology
  7. Notation
  8. Structure of the article
  9. Grassmannian, Fourier averages and switch lemmas
  10. Grassmannian
  11. Fourier averages and switch lemmas
  12. Almost Orthogonality for Directions in the Plane
  13. Algebraic almost orthogonality in general codimension
  14. Proof of Theorem \ref{['t:alldortho']}
  15. Sharp or nearly sharp bounds for maximal subspace averages
  16. ...and 14 more sections

Key Result

Theorem 1

For all $N>0$, $1\leq d < n$ there holds with an implicit numerical constant depending only upon $d,n$. This bound is sharp in terms of $N$ when $n=d+1$ and sharp up to the logarithmic factor in general.

Figures (1)

  • Figure 7.1: A plate $Q=P(I_Q,K_Q,v_Q)$ in $\mathbb{R}^3$ perpendicular to $v_Q\in V$.

Theorems & Definitions (62)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 2.2
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • ...and 52 more