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New files
On the Uncertainty Principle for Metaplectic Transformations,
(59 pages),
Journal of Functional Analysis,
Volume 289, Issue 5, 1 September 2025, 110997.
We explore the new proofs and extensions of the Heisenberg Uncertainty Principle introduced by A. Widgerson & Y. Widgerson in [MR4229152], developed in [MR4453622] by N.C. Dias, F. Luef and J.N. Prata and also in [MR4337266] by Y. Tang. In particular we give here a proof of the Uncertainty Principle for operators in the Metaplectic group in any dimension.
DOI number: https://doi.org/10.1016/j.jfa.2025.110997,
link on HAL.
On Some Singular Integrals Linked to Solvability and Signal Theory,
an article (pages 225 to 276)
in the book
Geometric Analysis of PDEs and Several Complex Variables
In Honor of Jorge Hounie's 75th Birthday,
edited by
S. Berhanu, N.Mir, G.Hoepfner,
published in 2024
in the Springer-Verlag Latin American Mathematics Series.
We explore in this paper the role of some singular integrals in the disproof of some estimates linked to solvability properties of pseudo-differential equations; it turns out that some of these singular integrals are also related to the construction of a counterexample to Flandrin's conjecture in signal theory.
DOI number: 10.1007/978-3-031-69702-9,
file on HAL.
Integrating the Wigner Distribution on subsets of the phase space, a Survey,
Memoirs of the European Mathematical Society, EMS Press,
volume 12, (1), (2024), 224 pages,
link on HAL.
We review several properties of integrals of the Wigner distribution on subsets of the phase space. Along our way, we provide a theoretical proof of the invalidity of Flandrin's conjecture, a fact already proven via numerical arguments in our joint paper [MR4054880]
with B. Delourme and T. Duyckaerts. We use also the J.G. Wood & A.J. Bracken paper [MR2131219],
for which we offer a mathematical perspective.
We review thoroughly the case of subsets of the plane whose boundary is a conic curve and show that Mehler's formula can be helpful in the analysis of these cases, including for the higher dimensional case investigated in the paper
[MR2761287] by
E. Lieb and Y. Ostrover.
Using the Feichtinger algebra, we show that, generically in the Baire sense,
the Wigner distribution of a pulse in
L2(ℝn)
does not belong to
L1(ℝ2n)
providing as a byproduct a large class of examples of subsets of the phase space
ℝ2n on which the integral of the Wigner distribution is infinite.
We study as well the case of convex polygons of the plane, with a rather weak estimate depending on the number of vertices, but independent of the area of the polygon.
The files video-São Carlos and
video-Reims (in French) might serve as an introduction to the topic.
On some properties of the curl operator and their consequences for the Navier-Stokes system
with François Vigneron,
Communications in Mathematical Research,
38 (2022), no.4, 449-497,
file on HAL,
file on arXiv,
March 2022.
We investigate some geometric properties of the curl operator, based on its diagonalization
and its expression as a non-local symmetry of the pseudo-derivative (-Δ)1/2 among divergence-free vector fields
with finite energy. In this context, we introduce the notion of spin-definite fields, i.e. eigenvectors of
(-Δ)-1/2 curl.
The two spin-definite components of a general 3D incompressible flow untangle the right-handed motion from the left-handed one.
Having observed that the non-linearity of Navier-Stokes has the structure of a cross-product
and its weak (distributional) form is a determinant that involves the vorticity, the velocity and a test function,
we revisit the conservation of energy and the balance of helicity in a geometrical fashion. We show that in the case
of a finite-time blow-up, both spin-definite components of the flow will explose simultaneously and with equal rates,
i.e. singularities in 3D are the result of a conflict of spin, which is impossible in the poorer geometry of 2D flows.
We investigate the role of the local and non-local determinants
∫ [0,T]
∫ ℝ3 det(curl u, u, (-Δ)θ u) dx dt
and their spin-definite counterparts, which drive the enstrophy and, more generally, are responsible for
the regularity of the flow and the emergence of singularities or quasi-singularities.
As such, they are at the core of turbulence phenomena.
An energy method for averaging lemmas,
with Diogo Arsénio,
Pure and Applied Analysis,
3 (2021), no. 2, 319-362,
file on arXiv, June 2020.
This work introduces a new approach to velocity averaging lemmas in kinetic theory. This approach---based upon the classical energy method---provides a powerful duality principle in kinetic transport equations which allows for a natural extension of classical averaging lemmas to previously unknown cases where the density and the source term belong to dual spaces. More generally, this kinetic duality principle produces regularity results where one can trade a loss of regularity or integrability somewhere in the kinetic transport equation for a suitable opposite gain elsewhere. Also, it looks simpler and more robust to rely on proving inequalities instead of constructing exact parametrices. The results in this article are introduced from a functional analytic point of view and they are motivated by the abstract regularity theory of kinetic transport equations.
updated
October 2, 2025
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