Open problems from Hörmander's 'Quadratic hyperbolic operators': direct microlocal analysis of the forward fundamental solution of the quadratic hyperbolic model

At the end of his 1991 lecture notes chapter 'Quadratic hyperbolic operators' (LNM 1495), Lars Hörmander posed five interlinked open questions about the forward fundamental solution $E$ of the model operator $Q(D_{n+1}x,D)+SD_{n+1}$ (equivalently $Q^{w}(x,D)+S$, the Weyl quantization of a real hyperbolic quadratic form $Q$ plus the constant subprincipal term $S$) that governs the local theory of hyperbolic partial differential operators with double characteristics: (1) prove that the twisted wave front set $WF'(E)$ contains the forward flowout of the characteristic diagonal (only the reverse containment being established in the chapter); (2) derive the subexponential growth estimate $\exp(\kappa\sqrt{d})$ of $E$ in the non-temperate case from energy estimates or microlocal methods alone, without the explicit Bessel-type formula; (3) determine the optimal weight in the growth estimates near the relative boundary of the dual cone $\Gamma^{\sigma}$; (4) prove by direct microlocal means (again without explicit formulas) that the fundamental solution is a Gevrey-4, respectively Gevrey-2, distribution in the two non-effectively hyperbolic normal-form cases; and (5) find a conceptual explanation for why the standard energy estimates yield only exponential growth in the effectively hyperbolic case while the fundamental solution is in fact temperate. Later work (Bernardi–Nishitani; Nishitani 2026) settled the general Gevrey well-posedness thresholds for these local models via the explicit formulas, but leaves the five direct, non-explicit problems below open.

Background

A second-order hyperbolic differential operator can fail to be strictly hyperbolic: its principal symbol may vanish together with its gradient at certain points, giving double characteristics. The local geometry governing well-posedness of the Cauchy problem there is captured, after blow-up (localization), by a model operator: the Weyl quantization $Q^{w}(x,D)+S$ of a real quadratic form $Q$ on phase space $\mathbb{R}^{2n}$ plus the value $S$ of the subprincipal symbol at the double point. The quadratic form is called hyperbolic with respect to a time direction $t$ if $Q$ vanishes on that direction and $Q<0$ on a punctured conic neighborhood of it. The associated Hamilton map $F$ (the fundamental matrix of $Q$, defined by $\sigma(X,FY)=Q(X,Y)$ with $\sigma$ the symplectic form) encodes the bicharacteristic flow linearized at the double point: if $F$ has real eigenvalues at every double characteristic point the operator is effectively hyperbolic and the Cauchy problem is well-posed in $C^{\infty}$ under the classical energy method; otherwise it is non-effectively hyperbolic, and $C^{\infty}$ well-posedness forces the Ivrii–Petkov–Hörmander condition $S+\sum_j \mu_j \ge 0$, where $\mu_j$ are the moduli of the purely imaginary eigenvalues of $F$ (the frequencies of harmonic-oscillator terms appearing in the normal forms). Well-posedness then holds only in Gevrey classes $G^s$ (functions whose derivatives grow like $(j!)^s$) below sharp thresholds. Hörmander's 1991 CIME chapter analyzed the forward fundamental solution $E$ of the model — the distribution kernel solving $(Q^{w}+S)E=\delta$ with support in forward time — by two complementary routes: microlocal propagation methods, giving an upper bound for the twisted wave front set $WF'(E)$ (the wave front set of the kernel as a distribution on the product, transported to the twisted cotangent variables $(x-y,\xi-\eta)$; here $WF'(E)$ is contained in the union of the diagonal, the forward flowout of the characteristic diagonal along the bicharacteristic flow, and exceptional sets in two degenerate cases where it is in fact determined exactly), and explicit computations: in the non-effectively hyperbolic normal forms (8.1) $Q=-\xi_0^2+\sum_{j\ge1}x_j^2$ and (8.2) $Q=-\xi_0^2+2\xi_0\xi_1+x_1^2+\sum_{j\ge2}x_j^2$ the kernel is given by Bessel $J_0$-type formulas, from which one reads off both the exact wave front set in the degenerate cases and sharp growth: $E$ is a temperate distribution (polynomially bounded) in the effectively hyperbolic case — already known from Hörmander 1977 — while in the non-effectively hyperbolic case it can grow subexponentially, $|E| \le C\exp(\kappa\sqrt{d})$ with $d$ the forward time separation, governed by the dual cone $\Gamma^{\sigma}$ (the component of $\{Q>0\}$ in the twisted variables containing the anti-time direction, whose boundary marks where growth estimates transition) and the set $\Lambda=\Gamma^{\sigma}\cap Q^{-1}(0)$. The chapter also introduced, as a byproduct for handling exponentially growing kernels, the Gabor wave front set and the space $\mathcal{S}'_{A}$ of exponentially growing distributions with the analytic wave front set $WF_A$ — tools that later took on a life of their own. The five closing questions ask for the parts of this picture that the explicit formulas deliver but the general microlocal machinery does not: they are the residual gap between the two routes of the chapter, and they remain the natural calibration points for how far energy-estimate and phase-space methods alone can go for weakly hyperbolic models.

Problem Statement

For the quadratic hyperbolic model operator $Q(D_{n+1}x,D)+SD_{n+1}$ (equivalently $Q^{w}(x,D)+S$), where $Q$ is a real quadratic form on phase space hyperbolic with respect to a time direction, with Hamilton map $F$ whose purely imaginary eigenvalue moduli are $\mu_j$, and $S$ the constant subprincipal term, let $E$ be the forward fundamental solution. Five problems from the closing remarks of Hörmander's 1991 chapter remain: (1) Singular structure: prove that the twisted wave front set $WF'(E)$ contains the forward flowout of the characteristic diagonal (the propagation set swept out by the forward bicharacteristic, i.e. Hamilton-flow, orbits issuing from the characteristic points of the diagonal) — the chapter proves only the reverse containment, together with an exact description in two degenerate cases ($F^{2}=0$; and the case where $F^{2}\neq0$ but $\Lambda$ is a distinguished Lagrangian subspace); a proof or counterexample in the general case is missing. (2) Growth without explicit formulas: in the non-effectively hyperbolic normal-form case (8.2), where $E$ is non-temperate, derive the subexponential growth estimate $|E|\le C\exp(\kappa\sqrt{d})$ (with $d=x_{0}-y_{0}>0$ the forward time separation) purely from energy estimates or microlocal methods, without using the explicit Bessel-type expression for the kernel. (3) Optimal weight: determine the optimal weight in the growth estimates near the relative boundary of the dual cone $\Gamma^{\sigma}$ (the chapter's estimates are known to be non-optimal there; it is only known that the weight falls off to become 1 at infinity outside any conic neighborhood of the cone). (4) Gevrey regularity by direct means: prove by direct microlocal methods, not relying on the explicit formulas, that the fundamental solution is a distribution on the Gevrey class 4, respectively 2, in the two non-effectively hyperbolic normal-form cases (8.2), respectively (8.1). (5) Temperate growth mechanism: find a conceptual explanation for why the energy estimates in the effectively hyperbolic case yield only exponential growth of the fundamental solution, while the actual growth is bounded/temperate (the temperate property itself being known from Hörmander 1977). The common objective of (1)–(5) is a microlocal, estimate-based theory of the forward fundamental solution of the quadratic hyperbolic model, matching what the explicit formulas deliver, and thereby explaining — rather than merely computing — the singularities, growth, and Gevrey regularity of $E$. The five parts are separately stated problems: resolving any one of them is a self-contained contribution, while resolving the task in full means an estimate-based account of all five. In particular, a resolution of (5) requires a rigorous argument, not a heuristic observation, that accounts for the exponential factor produced by the standard energy estimates and reconciles it with the temperateness of $E$.

References

1. Lars Hörmander, 'Quadratic hyperbolic operators', in: L. Cattabriga and L. Rodino (eds.), Microlocal Analysis and Applications, Lecture Notes in Mathematics 1495, Springer, Berlin, 1991, pp. 118–160. DOI: 10.1007/bfb0085123, https://doi.org/10.1007/bfb0085123.
2. Lars Hörmander, 'The Cauchy problem for differential equations with double characteristics', Journal d'Analyse Mathématique 32 (1977), 118–196. DOI: 10.1007/BF02803578, https://doi.org/10.1007/BF02803578 (temperate fundamental solution and effectively hyperbolic well-posedness).
3. Lars Hörmander, 'The wave front set of the fundamental solution of a hyperbolic operator with double characteristics', Journal d'Analyse Mathématique 59 (1992), 1–36. DOI: 10.1007/BF02790214, https://doi.org/10.1007/BF02790214 (constant-coefficient case only).
4. Lars Hörmander, 'Symplectic classification of quadratic forms, and general Mehler formulas', Mathematische Zeitschrift 219 (1995), 413–449. DOI: 10.1007/BF02572374, https://doi.org/10.1007/BF02572374 (systematizes the explicit-formula route).
5. Tatsuo Nishitani, Cauchy Problem for Differential Operators with Double Characteristics, Lecture Notes in Mathematics 2202, Springer, Cham, 2017. DOI: 10.1007/978-3-319-67612-8, https://doi.org/10.1007/978-3-319-67612-8.
6. Enrico Bernardi and Tatsuo Nishitani, 'On the Cauchy problem for noneffectively hyperbolic operators: The Gevrey 4 well-posedness', Kyoto Journal of Mathematics 51 (2011), 767–810. DOI: 10.1215/21562261-1424857, https://doi.org/10.1215/21562261-1424857.
7. Enrico Bernardi and Tatsuo Nishitani, 'On the Cauchy problem for non-effectively hyperbolic operators: The Gevrey 3 well-posedness', Journal of Hyperbolic Differential Equations 8 (2011), 615–650. DOI: 10.1142/S0219891611002512, https://doi.org/10.1142/S0219891611002512.
8. Tatsuo Nishitani, 'A more direct way to the Cauchy problem for effectively hyperbolic operators', Journal of Pseudo-Differential Operators and Applications 15, article 20 (2024). DOI: 10.1007/s11868-024-00592-4, https://doi.org/10.1007/s11868-024-00592-4 (open access).
9. Tatsuo Nishitani, 'A direct energy estimates for effectively hyperbolic operators', preprint (2021), arXiv:2108.03546, https://arxiv.org/abs/2108.03546 (simpler energy estimates for the effectively hyperbolic Cauchy problem).
10. Tatsuo Nishitani, 'A completion of our earlier work on the Cauchy problem for non-effectively hyperbolic operators', preprint (2026), arXiv:2602.23591, https://arxiv.org/abs/2602.23591.
11. Enrico Bernardi and Tatsuo Nishitani, 'Geometric results for hyperbolic operators with spectral transition of the Hamilton map', preprint (2025), arXiv:2505.21078, https://arxiv.org/abs/2505.21078.
12. Enrico Bernardi, 'Lack of Gevrey solvability for a model operator', preprint (2026), arXiv:2605.27054, https://arxiv.org/abs/2605.27054.
13. Enrico Bernardi and Leonardo Marconi, 'On some stochastic hyperbolic equations with symplectic characteristics', preprint (2024), arXiv:2409.01956, https://arxiv.org/abs/2409.01956 (explicit fundamental solutions for stochastically perturbed model hyperbolic operators).
14. Luigi Rodino and Patrik Wahlberg, 'The Gabor wave front set', Monatshefte für Mathematik 173 (2013), 625–655. DOI: 10.1007/s00605-013-0592-0, https://doi.org/10.1007/s00605-013-0592-0.
15. Karel Pravda-Starov, 'Generalized Mehler formula for time-dependent non-selfadjoint quadratic operators and propagation of singularities', Mathematische Annalen 372 (2018), 1335–1382. DOI: 10.1007/s00208-018-1667-y, https://doi.org/10.1007/s00208-018-1667-y (preprint arXiv:1703.02797, https://arxiv.org/abs/1703.02797).
16. S. Ivan Trapasso, 'Wave packet analysis of semigroups generated by quadratic differential operators', preprint (2024, revised 2025), arXiv:2408.11130, https://arxiv.org/abs/2408.11130.
17. Enrico Bernardi and Alberto Lanconelli, 'On a class of stochastic hyperbolic equations with double characteristics', preprint (2021), arXiv:2106.14715, https://arxiv.org/abs/2106.14715 (explicit fundamental solutions for stochastically perturbed model hyperbolic operators).