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Sharp Strichartz estimates on non-trapping asymptotically conic manifolds

Andrew Hassell, Terence Tao, Jared Wunsch

TL;DR

The paper proves local-in-time Strichartz estimates for Schrödinger evolution on non-trapping asymptotically conic manifolds, extending Euclidean-like dispersion to non-Euclidean geometries. It develops a robust framework of local Schrödinger integral operators (LSIOs) and a scattering calculus, introducing the X-space with half-angular smoothing to manage angular derivatives without derivative loss. A parametrix construction based on cones and balls, together with a positive-commutator argument, yields local smoothing and dispersive bounds; the Christ-Kiselev lemma then delivers the full non-endpoint Strichartz range for q>2. The analysis handles short-range and long-range metrics, including a detailed treatment of the phase Φ, geodesic structure, and conormal function algebras, culminating in a compositional LSIO calculus and dispersive estimates that feed into the Strichartz bounds and applications to semilinear Schrödinger equations. The results connect with prior work on asymptotically Euclidean and flat geometries, and suggest pathways to endpoint results and global-in-time theory.

Abstract

We obtain the Strichartz inequalities $$ \| u \|_{L^q_t L^r_x([0,1] \times M)} \leq C \| u(0) \|_{L^2(M)}$$ for any smooth $n$-dimensional Riemannian manifold $M$ which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and non-trapping, where $u$ is a solution to the Schrödinger equation $iu_t + {1/2} Δ_M u = 0$, and $2 < q, r \leq \infty$ are admissible Strichartz exponents ($\frac{2}{q} + \frac{n}{r} = \frac{n}{2}$). This corresponds with the estimates available for Euclidean space (except for the endpoint $(q,r) = (2, \frac{2n}{n-2})$ when $n > 2$). These estimates imply existence theorems for semi-linear Schrödinger equations on $M$, by adapting arguments from Cazenave and Weissler \cite{cwI} and Kato \cite{kato}. This result improves on our previous result in \cite{HTW}, which was an $L^4_{t,x}$ Strichartz estimate in three dimensions. It is closely related to the results of Staffilani-Tataru, Burq, Tataru, and Robbiano-Zuily, who consider the case of asymptotically flat manifolds.

Sharp Strichartz estimates on non-trapping asymptotically conic manifolds

TL;DR

The paper proves local-in-time Strichartz estimates for Schrödinger evolution on non-trapping asymptotically conic manifolds, extending Euclidean-like dispersion to non-Euclidean geometries. It develops a robust framework of local Schrödinger integral operators (LSIOs) and a scattering calculus, introducing the X-space with half-angular smoothing to manage angular derivatives without derivative loss. A parametrix construction based on cones and balls, together with a positive-commutator argument, yields local smoothing and dispersive bounds; the Christ-Kiselev lemma then delivers the full non-endpoint Strichartz range for q>2. The analysis handles short-range and long-range metrics, including a detailed treatment of the phase Φ, geodesic structure, and conormal function algebras, culminating in a compositional LSIO calculus and dispersive estimates that feed into the Strichartz bounds and applications to semilinear Schrödinger equations. The results connect with prior work on asymptotically Euclidean and flat geometries, and suggest pathways to endpoint results and global-in-time theory.

Abstract

We obtain the Strichartz inequalities for any smooth -dimensional Riemannian manifold which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and non-trapping, where is a solution to the Schrödinger equation , and are admissible Strichartz exponents (). This corresponds with the estimates available for Euclidean space (except for the endpoint when ). These estimates imply existence theorems for semi-linear Schrödinger equations on , by adapting arguments from Cazenave and Weissler \cite{cwI} and Kato \cite{kato}. This result improves on our previous result in \cite{HTW}, which was an Strichartz estimate in three dimensions. It is closely related to the results of Staffilani-Tataru, Burq, Tataru, and Robbiano-Zuily, who consider the case of asymptotically flat manifolds.

Paper Structure

This paper contains 21 sections, 21 theorems, 233 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The geometry of asymptotically conic manifolds
  3. Local Schrödinger integral operators (LSIOs)
  4. Scattering symbol calculus and the space $X$
  5. Construction of the parametrix
  6. Proof of Theorem \ref{['main']}
  7. Remarks
  8. Appendix: Christ-Kiselev lemma
  9. Appendix: Phase nondegeneracy estimates
  10. Proof of Lemma \ref{['glc']}
  11. Proof of Lemma \ref{['grad1']}
  12. Proof of Lemma \ref{['geodesic-lemma']}
  13. Appendix: The geometry of long range metrics
  14. The blown up product manifold $\overline{M}^2_b$.
  15. Algebras of conormal functions
  16. ...and 6 more sections

Key Result

Theorem 1.3

For any $n$-dimensional asymptotically conic non-trapping manifold $M$, with either short range or long range metric $g$, and any (Schwartz) solution $u$ to nls, we have for any admissible pair $(q,r)$ with $q > 2$.

Figures (2)

  • Figure 1: The radial compactification map
  • Figure 2: The blown up space $\overline{M}^2_b$

Theorems & Definitions (52)

  • Remark 1.1
  • Example 1.2
  • Theorem 1.3: Strichartz estimate
  • Remark 1.4
  • Remark 1.5
  • Lemma 1.6: Half-angular local smoothing effect HTW
  • Lemma 2.1: Geodesic local connectedness of $\overline{M}$
  • proof
  • Remark 2.2
  • Example 2.3
  • ...and 42 more