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Angular Regularity and Strichartz Estimates for the Wave Equation

Jacob Sterbenz, Igor Rodnianski

TL;DR

The paper addresses sharp Strichartz-type estimates for the linear wave equation on Minkowski space under additional angular regularity. It develops a framework combining spherical harmonic analysis, Hankel transforms, and wave-packet phase-space localization to substantially extend admissible (q,r) ranges beyond classical results. Two proofs are given, including a Hankel-φ approach and an angular-energy method, with extensions to multilinear estimates; an appendix provides a Rodnianski-style endpoint proof in low dimensions. These results enhance our ability to treat nonlinear wave equations lacking null structures, enabling global existence and scattering results for models such as Yang–Mills in Lorentz gauge for suitable angular-regular initial data.

Abstract

We prove here essentially sharp linear and bilinear Strichartz type estimates for the wave equations on Minkowski space, where we assume the initial data possesses additional regularity with respect to fractional powers of the usual angular momentum operators. In this setting, the range of (q,r) exponents vastly improves over what is available for the wave equations based on translation invariant derivatives of the initial data and the dispersive inequality. Two proofs of this result are given.

Angular Regularity and Strichartz Estimates for the Wave Equation

TL;DR

The paper addresses sharp Strichartz-type estimates for the linear wave equation on Minkowski space under additional angular regularity. It develops a framework combining spherical harmonic analysis, Hankel transforms, and wave-packet phase-space localization to substantially extend admissible (q,r) ranges beyond classical results. Two proofs are given, including a Hankel-φ approach and an angular-energy method, with extensions to multilinear estimates; an appendix provides a Rodnianski-style endpoint proof in low dimensions. These results enhance our ability to treat nonlinear wave equations lacking null structures, enabling global existence and scattering results for models such as Yang–Mills in Lorentz gauge for suitable angular-regular initial data.

Abstract

We prove here essentially sharp linear and bilinear Strichartz type estimates for the wave equations on Minkowski space, where we assume the initial data possesses additional regularity with respect to fractional powers of the usual angular momentum operators. In this setting, the range of (q,r) exponents vastly improves over what is available for the wave equations based on translation invariant derivatives of the initial data and the dispersive inequality. Two proofs of this result are given.

Paper Structure

This paper contains 9 sections, 13 theorems, 178 equations.

Table of Contents

  1. Introduction: Classical Strichartz estimates, improvements for spherically symmetric data, and Knapp counterexamples
  2. Basic notation
  3. Some results from analysis on the sphere.
  4. Reduction of Theorem \ref{['unit_ang_str_th']} to a frequency localized endpoint
  5. The Hankel transform
  6. The Hankel--$\varphi$ transform
  7. Some $L^2$ dispersive estimates for the wave equation; linear and bilinear estimates
  8. Bilinear estimates for angularly regular data
  9. Appendix

Key Result

Theorem 1.1

Let $3 \leqslant n$ be the number of spatial dimensions, and let $\sigma=\frac{n-1}{2}$, then the following estimate holds for $2\leqslant q$: where $\frac{1}{q} + \frac{\sigma}{r} \leqslant \frac{\sigma}{2}$, with the exception of the forbidden $L^2(L^\infty)$ endpoint on $\mathbb{R}^{3+1}$.

Theorems & Definitions (21)

  • Theorem 1.1: "Classical" Strichartz estimates including endpoints (see KT_str, GB, and ST_classical)
  • Proposition 1.2: Unit frequency Strichartz estimates for spherically symmetric data
  • Theorem 1.3: Strichartz estimates for spherically symmetric initial data
  • Remark 1.4
  • Theorem 1.5: Strichartz estimates for angularly regular data
  • Remark 1.6
  • Remark 1.7
  • Lemma 3.1: Basic properties of spherical harmonics (see e.g. SWeuclid
  • Proposition 3.2: Interpolation of the angular Sobolev spaces $H^s_\Omega$
  • Proposition 3.3: Littlewood--Paley--Stein theorem for the sphere (see Ssi, Slp, and STRlp)
  • ...and 11 more