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Oscillatory integral operators on manifolds and related Kakeya and Nikodym problems

Song Dai, Liuwei Gong, Shaoming Guo, Ruixiang Zhang

Abstract

We consider Carleson-Sjölin operators on Riemannian manifolds that arise naturally from the study of Bochner-Riesz problems on manifolds. They are special cases of Hörmander-type oscillatory integral operators. We obtain improved $L^p$ bounds of Carleson-Sjölin operators in two cases: The case where the underlying manifold has constant sectional curvature and the case where the manifold satisfies Sogge's chaotic curvature condition. The two results rely on very different methods: To prove the former result, we show that on a Riemannian manifold, the distance function satisfies Bourgain's condition if and only if the manifold has constant sectional curvature. To obtain the second result, we introduce the notion of "contact orders" to Hörmander-type oscillatory integral operators, prove that if a Hörmander-type oscillatory integral operator is of a finite contact order, then it always has better $L^p$ bounds than "worst cases" (in spirit of Bourgain and Guth, and Guth, Hickman and Iliopoulou), and eventually verify that for Riemannian manifolds that satisfy Sogge's chaotic curvature condition, their distance functions alway have finite contact orders. As byproducts, we obtain new bounds for Nikodym maximal functions on manifolds of constant sectional curvatures.

Oscillatory integral operators on manifolds and related Kakeya and Nikodym problems

Abstract

We consider Carleson-Sjölin operators on Riemannian manifolds that arise naturally from the study of Bochner-Riesz problems on manifolds. They are special cases of Hörmander-type oscillatory integral operators. We obtain improved bounds of Carleson-Sjölin operators in two cases: The case where the underlying manifold has constant sectional curvature and the case where the manifold satisfies Sogge's chaotic curvature condition. The two results rely on very different methods: To prove the former result, we show that on a Riemannian manifold, the distance function satisfies Bourgain's condition if and only if the manifold has constant sectional curvature. To obtain the second result, we introduce the notion of "contact orders" to Hörmander-type oscillatory integral operators, prove that if a Hörmander-type oscillatory integral operator is of a finite contact order, then it always has better bounds than "worst cases" (in spirit of Bourgain and Guth, and Guth, Hickman and Iliopoulou), and eventually verify that for Riemannian manifolds that satisfy Sogge's chaotic curvature condition, their distance functions alway have finite contact orders. As byproducts, we obtain new bounds for Nikodym maximal functions on manifolds of constant sectional curvatures.
Paper Structure (21 sections, 25 theorems, 374 equations)

This paper contains 21 sections, 25 theorems, 374 equations.

Table of Contents

  1. Introduction
  2. Hörmander-type oscillatory integrals
  3. Curved Kakeya problem
  4. Carleson-Sjölin operators on manifolds
  5. Nikodym problems on manifolds
  6. Statement of main results
  7. Proof of Theorem \ref{['230711thm2_5']}
  8. Preliminaries in Riemannian geometry
  9. Sogge's chaotic curvature condition
  10. Proof of Theorem \ref{['230816theorem2_4']}
  11. Preliminaries in polynomial partitionings
  12. Polynomial partitioning algorithms
  13. Polynomial Wolff axioms
  14. Proofs of Theorem \ref{['230617thm2_1']} and Theorem \ref{['230323theorem3_4']}
  15. Preliminaries in Riemannian geometry II
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\phi$ be a phase function of the form 230324e1_4. Let $s_0$ be the signature of the matrix $A$. Then 230324e1_5 holds for all

Theorems & Definitions (57)

  • Theorem 1.1: Hickman and Iliopoulou, HI22
  • Theorem 1.2: Stein Ste84, Bourgain and Guth BG11
  • Theorem 1.3: Lee Lee06; Guth, Hickman, Iliopoulou, GHI19
  • Definition 1.4: Bourgain's condition, Bou91, GWZ22
  • Theorem 1.5: Guo, Wang and Zhang GWZ22
  • Definition 1.6: Curved tubes
  • Definition 1.7: Curved Kakeya sets
  • Definition 1.8: Curved Kakeya maximal function, Bou91
  • Theorem 1.9: Wisewell Wis05
  • Definition 1.10: Nikodym set, Sogge Sog99
  • ...and 47 more