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Spherical maximal functions and Hardy spaces for Fourier integral operators

Abhishek Ghosh, Naijia Liu, Jan Rozendaal, Liang Song

Abstract

We use the Hardy spaces for Fourier integral operators to obtain bounds for spherical maximal functions in $L^{p}(\mathbb{R}^{n})$, $n\geq2$, where the radii of the spheres are restricted to a compact interval in $(0,\infty)$. These bounds extend to general hypersurfaces with non-vanishing Gaussian curvature, to the complex spherical means, and to geodesic spheres on compact manifolds. We also obtain improved maximal function bounds and pointwise convergence statements for wave equations, both on $\mathbb{R}^{n}$ and on compact manifolds. The maximal function bounds are essentially sharp for all $p\in[1,2]\cup [\frac{2(n+1)}{n-1},\infty)$, for each such hypersurface, every complex spherical mean, and on every manifold.

Spherical maximal functions and Hardy spaces for Fourier integral operators

Abstract

We use the Hardy spaces for Fourier integral operators to obtain bounds for spherical maximal functions in , , where the radii of the spheres are restricted to a compact interval in . These bounds extend to general hypersurfaces with non-vanishing Gaussian curvature, to the complex spherical means, and to geodesic spheres on compact manifolds. We also obtain improved maximal function bounds and pointwise convergence statements for wave equations, both on and on compact manifolds. The maximal function bounds are essentially sharp for all , for each such hypersurface, every complex spherical mean, and on every manifold.
Paper Structure (27 sections, 21 theorems, 155 equations)

This paper contains 27 sections, 21 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Previous work
  4. Discussion of the results
  5. Ideas behind the proofs
  6. Organization
  7. Notation and terminology
  8. Hardy spaces for Fourier integral operators
  9. Definitions on $\mathbb{R}^{n}$
  10. Results on $\mathbb{R}^{n}$
  11. Spaces on manifolds
  12. Maximal theorems
  13. Fourier integral operators
  14. The low frequencies
  15. The high frequencies
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $p\in[1,\infty)$ and $s>d(p)+\frac{1}{p}-\frac{n-1}{2}$. Then there exists a $C\geq0$ such that for all $f\in C(\mathbb{R}^{n})\cap \mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$. Conversely, if $p\in[1,2]\cup [\frac{2(n+1)}{n-1},\infty)$ and eq:maximalhyp holds for all $f\in\mathcal{S}(\mathbb{R}^{n})$, then $s\geq d(p)+\frac{1}{p}-\frac{n-1}{2}$.

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 47 more