Endpoint Strichartz Estimates

• Endpoint Strichartz estimates are precise inequalities that determine the maximal spacetime integrability for solutions to dispersive linear equations like the Schrödinger and wave equations.
• They are established using TT* methods, interpolation, and microlocal analysis, and extend to various settings including non-trapping and conical geometries.
• These estimates are vital for nonlinear analysis, influencing key aspects such as well-posedness, scattering, and regularity thresholds in diverse PDE models.

Endpoint Strichartz estimates are a central component of the theory of dispersive partial differential equations, characterizing the limiting spacetime integrability properties of solutions to linear evolution equations such as the Schrödinger and wave equations. The endpoint refers to those pairs of exponents at which the Strichartz inequalities reach critical scaling or summability, often marking the boundary of what is attainable by the TT* and interpolation techniques. The endpoint case is both structurally subtle and crucial for nonlinear analysis, as it governs threshold regularity for well-posedness, scattering, and critical nonlinear interactions.

1. Definition and General Theory

Given a linear dispersive evolution (e.g., the free Schrödinger equation $i\partial_t u + \Delta u = 0$), the homogeneous Strichartz estimates are inequalities of the form

$$\|e^{it\Delta} f\|_{L^p_t L^q_x(\mathbb{R}^{1+n})} \leq C \|f\|_{L^2_x(\mathbb{R}^n)}$$

where the pair $(p,q)$ is admissible if $2 \leq p, q \leq \infty$, $2/p + n/q = n/2$, and $(p,q,n)\neq (2,\infty,2)$. The endpoint refers to the case $(p,q) = (2, 2n/(n-2))$ when $n > 2$, providing maximal integrability compatible with scaling and dispersive decay (Frank et al., 2014, Mizutani, 2016).

Endpoint Strichartz estimates are also relevant in the context of wave, kinetic transport, and other dispersive models, as well as in the analysis of nontrivial geometries (e.g., exterior domains, conic manifolds, hyperbolic spaces) and flows with additional structure (e.g., magnetic or inverse-square potentials) (Hassell et al., 2013, Georgiev et al., 2019, Li, 2017).

2. Endpoint Strichartz Estimates: Free and Geometric Settings

In Euclidean space for $n\geq 3$, the endpoint estimate is valid for the Schrödinger propagator: $$\|e^{it\Delta} f\|_{L^2_t L^{2n/(n-2)}_x} \leq C\|f\|_{L^2_x}$$ and similarly for the inhomogeneous Duhamel term (Frank et al., 2014).

These estimates extend to variable-coefficient and geometric settings under suitable nontrapping and regularity conditions:

• Non-trapping asymptotically conic manifolds: Endpoint Strichartz estimates (homogeneous and inhomogeneous) hold for the Laplace-Beltrami operator plus smooth potentials, with full admissibility and no loss of derivatives (Hassell et al., 2013).
• Exterior domains: Under the nontrapping condition for obstacles, homogeneous and inhomogeneous endpoint Strichartz inequalities are valid for the Dirichlet Laplacian (Georgiev et al., 2019).
• Conical manifolds: Sharp endpoint estimates for Schrödinger equations with angular curvature-dependent Laplacians, with the endpoint bound depending on the smallest eigenvalue of an associated operator on the cross-section (Zhang et al., 2017).
• Elastic and damped wave equations: Endpoint (wave-admissible) Strichartz inequalities have been established through careful symbol diagonalization and dispersive analysis, even with additional damping or coupling (Kim et al., 2021, Inui et al., 2019).
• Magnetic and inverse-square potentials: The endpoint may degrade to a weak-type (Lorentz space) form in the presence of critical potentials (Mizutani, 2016).

A representative table of endpoint Strichartz results in selected geometric settings:

Problem/Operator	Endpoint Estimate Valid?	Remarks
Free Schrödinger ($n\geq 3$)	Yes, $(2,2n/(n-2))$	Strong-type
Schrödinger + subcritical $|x|^{-2}$	Yes, full range by dispersion	Same as free case
Schrödinger + critical $|x|^{-2}$	Yes for nonradial, weak (Lorentz) for radial	Strong/weak splitting (Mizutani, 2016)
Non-trapping exterior domain	Yes, $(2,2n/(n-2))$	Requires non-trapping (Georgiev et al., 2019)
Conical manifold	Yes, $(2,2n/(n-2))$	Range influenced by lowest eigenvalue
Hyperbolic/magnetic wave eq.	Yes (with weights/derivatives in some cases)	Weighted estimates often crucial

3. Lorentz Spaces, Weak-Type Endpoints, and Spherical Averages

The endpoint may not always be attainable in strong Lebesgue norms. For critical potentials or in low dimensions, the best possible result may be a Lorentz-space or spherically averaged estimate:

• Critical inverse-square potential: The endpoint for the radial component is only available in weak-type Lorentz space $L^2_t L^{2n/(n-2),\infty}_x$, with strong-type failing (explicit radial counterexamples) (Mizutani, 2016).
• 2D Schrödinger equation: The $(2,\infty)$ endpoint fails; Tao showed that a spherically averaged endpoint of the form $$\|e^{it\Delta} f\|_{L^2_t L^\infty_r L^2_\omega}\lesssim \|f\|_{L^2}$$ is available (Kim et al., 2019).

Techniques to recover endpoint-type control include:

• Averaging in spherical coordinates or adding angular derivatives;
• Weakening to Lorentz or mixed-norm spaces;
• Imposing radial symmetry or angular regularity (Schippa, 2016, Cho et al., 2012).

4. Endpoint Boundaries for Orthonormal Systems and Trace Ideals

In problems involving orthonormal systems (quantum many-body, mean-field), endpoint Strichartz estimates control quantities like

$$\Bigl\|\sum_j \nu_j |e^{it\Delta} f_j|^2\Bigr\|_{L^r_{t,x}} \leq C \left(\sum_j |\nu_j|^{r'}\right)^{1/r'}$$

where $f_j$ is an orthonormal system and exponents must satisfy fine summability dictated by trace ideal interpolation (Frank et al., 2014). The endpoint again emerges as the boundary of allowable exponents, and may fail in the strongest form, especially for spectral flows with additional structure (e.g., special Hermite operator (Ghosh et al., 21 Nov 2025)).

Failure of strong-type endpoint inequalities for orthonormal families may manifest as a loss of locality, with restricted weak-type or frequency-localized bounds remaining valid (Bez et al., 2017).

5. Failure of Endpoint Strichartz in Transport and Related Models

A notable counterexample comes from the kinetic transport equation: $\partial_t f + v\cdot\nabla_x f = 0$ where Strichartz-type mixed-norm estimates fail at the critical scaling endpoint in all dimensions (Bennett et al., 2013). This contrasts starkly with the Schrödinger case, highlighting the need for dispersive decay to achieve endpoint Strichartz estimates. Multilinear analysis recovers non-endpoint ranges.

6. Technical Overview: Sharp Methods for Endpoint Attainment

Common methodologies for establishing endpoint Strichartz estimates include:

• Microlocal analysis: Partition of unity in phase-space and microlocal parametrix constructions, especially in non-Euclidean geometries (Hassell et al., 2013, Zhang et al., 2017).
• TT*, bilinear/dual approaches: Keel–Tao abstract theorem and its bilinear refinements handle sharp interpolation and exploit dispersive decay (Frank et al., 2014).
• Resolvent and smoothing estimates: Kato smoothing and resolvent bounds enable endpoint control under perturbations, trapping, or singular geometries (Li, 2017, Mizutani, 2016, Kim et al., 2021).
• Spectral multiplier and interpolation in Schatten spaces: For orthonormal systems, the endpoint is governed by interpolation in trace ideals and duality with Schatten–class bounds (Frank et al., 2014, Ghosh et al., 21 Nov 2025).

7. Applications and Limitations

Endpoint Strichartz estimates are essential for:

• Critical well-posedness and scattering for nonlinear Schrödinger equations, including in conic or exterior geometries (Zhang et al., 2017, Georgiev et al., 2019);
• Unconditional uniqueness in critical energy spaces for damped or structurally perturbed wave equations (Inui et al., 2019);
• Fock space analysis of waves in random media, with endpoint estimates driving fixed-point arguments and large-time propagation (Breteaux et al., 2022).

Their limitations include:

• Failure at the endpoint in transport, certain orthonormal, or critical radial cases;
• Necessity of angular regularity or Lorentz space relaxation in low dimensions or with underlying anisotropy (Schippa, 2016, Mizutani, 2016);
• Sharp geometric and spectral assumptions (nontrapping, nonresonance) for full endpoint validity (Hassell et al., 2013, Li, 2017).

In summary, endpoint Strichartz estimates delineate the frontier of space-time integrability for dispersion-governed flows, their reach and form encoding deep connections between PDE symmetries, geometry, and harmonic analysis. Their appearance and failure demarcate the boundary between critical linear control and nonlinear regularity thresholds across a spectrum of models and settings.