Publications

We investigate dispersive and Strichartz estimates for the Schr\"odinger equation involving the fractional Laplacian in real hyperbolic spaces and their discrete analogues, homogeneous trees. Due to the Knapp phenomenon, the Strichartz estimates on Euclidean spaces for the fractional Laplacian exhibit loss of derivatives. A similar phenomenon appears on real hyperbolic spaces. However, such a loss disappears on homogeneous trees, due to the triviality of the estimates for small times.

We prove that the Patterson-Sullivan and Wigner distributions on the unit sphere bundle of a convex-cocompact hyperbolic surface are asymptotically identical. This generalizes results in the compact case by Anantharaman-Zelditch and Hansen-Hilgert-Schröder.

Already back in the 17-18th centuries, important foundations of modern statistical theory were formulated with the goal to address astronomical problems. This successful interdisciplinary collaboration has been revived since the 1990s, giving rise to the research flow called astrostatistics, which has been particularly active over the past decades. The increasing amount of astronomical data nowadays has posed new challenges and created the need for more innovative modern statistical theories and models. Directional statistics, a branch of statistics involving observations such as directions, axes, rotations, with values on (compact) Riemannian manifolds like our celestial sphere, has proved to be a promising domain to address important space sciences issues such as space weather, cosmology, or even space surveillance.
In this paper, we will instigate directional statistics by providing a review of their old and recent developments stimulated by interesting applications in space sciences.

The description of the Paley–Wiener space for compactly supported smooth functions $C^\infty_c(G)$ on a semi-simple Lie group $G$ involves certain intertwining conditions that are difficult to handle. In the present paper, we make them completely explicit for $G=\mathrm{SL}(2,\mathbb{R})^d$ $(d \in \mathbb{N})$ and $G=\mathrm{SL}(2,\mathbb{C})$. Our results are based on a defining criterion for the Paley–Wiener space, valid for general groups of real rank one, that we derive from Delorme’s proof of the Paley–Wiener theorem. In a forthcoming paper, we will show how these results can be used to study solvability of invariant differential operators between sections of homogeneous vector bundles over the corresponding symmetric spaces.

We study the Fourier transform for compactly supported distributional sections of complex homogeneous vector bundles on symmetric spaces of non-compact type $X=G/K$. We prove a characterisation of their range. In fact, from Delorme's Paley-Wiener theorem for compactly supported smooth functions on a real reductive group of Harish-Chandra class, we deduce topological Paley-Wiener and Paley-Wiener-Schwartz theorems for sections.

We show how the Fourier transform for distributional sections of vector bundles over symmetric spaces of non-compact type $G/K$ can be used for questions of solvability of systems of invariant differential equations in analogy to Hörmander's proof of the Ehrenpreis-Malgrange theorem. We get complete solvability for the hyperbolic plane $\mathbb{H}^2$ and partial results for products $\mathbb{H}^2 \times \cdots \times \mathbb{H}^2$ and the hyperbolic 3-space $\mathbb{H}^3$.

Monitoring plantar load conditions becomes useful in many health care fields, e.g. podiatric and orthopedic applications, rehabilitation tools, sports and fitness training tools, and in-field diagnosis and prevention tools for posture, balance, loading and contact times monitoring. IEE target is to provide a single insole-solution for daily usage in order to acquire information on the plantar load distribution for health prophylaxis in a large range of different shoe configurations. In this paper, we introduce for the first time a new High-Dynamic (HD) multi-cell smart insole sensor enabling advanced real-time foot plantar pressure monitoring applications. The in-situ measurement of the dynamic plantar load distribution provides an important new source of information that can be combined with traditional monitoring systems often based on accelerometer and gyroscope sensors. In fact, the new smart insole as presented here, facilitates the discovery in an early phase of any biomechanical mismatch in the walking or running gait of its user. Specific datasets have been recorded from a representative healthy population with different monitoring tools, i.e. force plate, pressure matrix and our new smart insole. The aim was to study the similarity of measurements recorded by each system on a defined measurement protocol. It is shown that the new monitoring device provides a competitive methodology to measure static and dynamic foot plantar pressure distribution. The system flexibility and robustness enable the development of new real-time applications, such as high peak pressure detection for diabetics, activity tracking, etc.