Research
I develop sharp estimates and microlocal tools for sub-elliptic operators: operators, such as sub-Laplacians on sub-Riemannian manifolds, which are not elliptic but are hypoelliptic under geometric conditions. For these operators the phase space carries a noncommutative group structure, characteristic varieties are singular, and dispersive behavior differs fundamentally from the elliptic case. My work combines microlocal and semiclassical analysis with noncommutative harmonic analysis and representation theory. Current directions:
Strichartz Estimates and Fourier Restriction for Sub-Laplacians
- With Davide Barilari, I proved refined Strichartz estimates for the Schrödinger and wave equations on H-type groups, with no symmetry assumptions on the initial data (arXiv:2501.04415). The main tool is a Fourier restriction theorem for the Heisenberg fan, the singular characteristic variety of the sub-Laplacian.
- Sharp constants, extremizers, and stability for restriction inequalities in the sub-Riemannian setting.
Microlocal Analysis on Filtered Manifolds
- With Clotilde Fermanian-Kammerer and Véronique Fischer, I constructed a pseudodifferential calculus on filtered manifolds, quantizing operator-valued symbols in local adapted frames (arXiv:2412.17448, accepted at Memoirs of the European Mathematical Society). The calculus admits parametrices for Hörmander sums of squares.
- Quantum limits of sub-elliptic Schrödinger equations: Egorov-type theorems and semiclassical measures on step-2 nilmanifolds, and the correct notion of classical flow on a filtered manifold.
Spectral Asymptotics and Eigenfunction Estimates
- Whether the spectral function of a Schrödinger operator with smooth bounded potential admits a complete pointwise asymptotic expansion, a question raised by Galkowski–Parnovski–Shterenberg. I am constructing potentials and delocalized quasimodes towards a counterexample.
- Sup-norm bounds for eigenfunctions of sub-Laplacians on contact manifolds, with the exponent (n−1)/2 in Hörmander's bound replaced by (Q−1)/2, where Q is the homogeneous dimension. This requires extending geodesic beam techniques to the sub-Riemannian setting.
Integral Geometry and Inverse Problems
- Injectivity, stability, and Fourier slice theorems for geodesic X-ray transforms on the Heisenberg group (J. Funct. Anal., 2021) and on H-type groups (arXiv:2312.00594).
- With François Monard, scattering rigidity for Yang–Mills fields on surfaces: recovering a connection from the scattering relation of the associated particle dynamics.