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Problems on spherical maximal functions

Joris Roos, Andreas Seeger

TL;DR

This survey synthesizes the theory of spherical maximal functions when radii are restricted to fractal dilation sets, connecting lacunary and general dilation regimes with L^p bounds, weighted inequalities, and sparse domination. It develops a cohesive framework linking L^p improving ranges to geometric fractal dimensions (Minkowski, Assouad, and their spectra) and introduces the Legendre–Assouad transform nu_E^sharp to describe radial phenomena, all while highlighting open endpoint questions. The work extends to multiparameter and non-Euclidean settings (helical averages, Nevo–Thangavelu means) and formulates a fractal local smoothing conjecture that generalizes Sogge-style smoothing to fractal radii, with deep connections to decoupling, Strichartz estimates, and fractal geometry. The results provide a unifying lens for a broad class of generalized Radon-type transforms and offer precise conjectural thresholds, guiding future research on a.e. convergence, weighted bounds, and endpoint behavior in fractal geometries.

Abstract

We survey old and new conjectures and results on various types of spherical maximal functions, emphasizing problems with a fractal dilation set.

Problems on spherical maximal functions

TL;DR

This survey synthesizes the theory of spherical maximal functions when radii are restricted to fractal dilation sets, connecting lacunary and general dilation regimes with L^p bounds, weighted inequalities, and sparse domination. It develops a cohesive framework linking L^p improving ranges to geometric fractal dimensions (Minkowski, Assouad, and their spectra) and introduces the Legendre–Assouad transform nu_E^sharp to describe radial phenomena, all while highlighting open endpoint questions. The work extends to multiparameter and non-Euclidean settings (helical averages, Nevo–Thangavelu means) and formulates a fractal local smoothing conjecture that generalizes Sogge-style smoothing to fractal radii, with deep connections to decoupling, Strichartz estimates, and fractal geometry. The results provide a unifying lens for a broad class of generalized Radon-type transforms and offer precise conjectural thresholds, guiding future research on a.e. convergence, weighted bounds, and endpoint behavior in fractal geometries.

Abstract

We survey old and new conjectures and results on various types of spherical maximal functions, emphasizing problems with a fractal dilation set.

Paper Structure

This paper contains 10 sections, 8 theorems, 67 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Lacunary dilation sets
  3. General dilation sets
  4. Weighted norm inequalities and sparse domination
  5. Lp improving
  6. Maximal functions: Further directions
  7. Multiparameter spherical maximal functions
  8. Helical averages
  9. The Nevo-Thangavelu spherical means
  10. A fractal local smoothing conjecture

Key Result

Theorem 3.1

SeegerWaingerWright1995 Let $\beta={\mathrm{Dim}}_{\mathrm{M}}{{\mathcal{E}}}$ and $p_{\mathrm{cr}}(\beta) = 1+\frac{\beta}{d-1}$. Then ${{\mathcal{M}}}_{{\mathcal{E}}}$ is bounded on $L^p$ for $p>p_{\mathrm{cr}}(\beta)$ and unbounded on $L^p$ for $p<p_{\mathrm{cr}}(\beta)$.

Figures (2)

  • Figure 1: Possible type sets.
  • Figure 2: Building a type set.

Theorems & Definitions (18)

  • Remark 2.2
  • Theorem 3.1
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • Theorem 4.2
  • Remark 4.4
  • Remark 4.5
  • Theorem 5.2
  • Theorem 5.3
  • ...and 8 more