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Makhlouf Derridj

Publications and source records attributed to Makhlouf Derridj.

8 recordsLinked to original sources
arXiv2024

On the Microlocal Regularity of the Gevrey Vectors for second order partial differential operators with non negative characteristic form of first kind

We study the microlocal regularity of the analytic/Gevrey vectors for the following class of second order partial differential equations \begin{align*} P(x,D) = \sum_{\ell,j=1}^{n} a_{\ell,j}(x) D_{\ell} D_{j} + \sum_{\ell=1}^{n} i b_{\ell}(x) D_{\ell} +c(x), \end{align*} where $a_{\ell,j}(x) = a_{j,\ell}(x)$, $b_{\ell}(x)$, $\ell,j \in \lbrace 1,\dots,\, n\rbrace$, are real valued real Gevrey functions of order $s$ and $c(x)$ is a Gevrey function of order $s$, $s \geq 1$, on $Ω$ open neighborhood of the origin in $\mathbb{R}^{n}$. Thus providing a microlocal version of a result due to M. Derridj in "Gevrey regularity of Gevrey vectors of second order partial differential operators with non negative characteristic form", Complex Anal. Synerg. $\mathbf{6}$, 10 (2020), https://doi.org/10.1007/s40627-020-00047-8.

math.AP↗
arXiv2022

On the microlocal regularity of the analytic vectors for "sums of squares" of vector fields

We prove via FBI-transform a result concerning the microlocal Gevrey regularity of analytic vectors for operators sums of squares of vector fields with real-valued real analytic coefficients of Hörmander type, thus providing a microlocal version, in the analytic category, of a result due to M. Derridj in "Local estimates for Hörmander's operators of first kind with analytic Gevrey coefficients and application to the regularity of their Gevrey vectors", concerning the problem of the local regularity for the Gevrey vectors for sums of squares of vector fields with real-valued real analytic/Gevrey coefficients. Nous démontrons , en utilisant la transformation de Fourier-Bros-Iagolnitzer, un résultat de régularité Gevrey microlocale , optimale, des vecteurs analytiques d'opérateurs de Hörmander de type "Sommes de carrés de champs de vecteurs" à coefficients analytiques sur un ouvert. Ce résultat est, dans le cadre analytique, la version microlocale du résultat de M.Derridj "Local estimates for Hörmander's operators of first kind with analytic Gevrey coefficients and application to the regularity of their Gevrey vectors", obtenu pour les vecteurs de Gevrey de tels opérateurs à coefficients Gevrey.

math.AP↗
arXiv2006

Subelliptic estimates for some systems of complex vector fields : quasihomogeneous case

For about twenty five years it was a kind of folk theorem that complex vector-fields defined on $Ω\times \mathbb R_t$ (with $Ω$ open set in $\mathbb R^n$) by $$ L_j = \frac{\partial}{\partial t_j} + i \frac {\partial ϕ}{\partial t_j}(\t) \frac{\partial}{\partial x}, j=1,..., n, \t\in Ω, x\in \mathbb R ,$$ with $ϕ$ analytic, were subelliptic as soon as they were hypoelliptic. This was the case when $n=1$ but in the case $n>1$, an inaccurate reading of the proof given by Maire (see also Trèves) of the hypoellipticity of such systems, under the condition that $ϕ$ does not admit any local maximum or minimum (through a non standard subelliptic estimate), was supporting the belief for this folk theorem. Quite recently, J.L. Journé and J.M.Trépreau show by examples that there are very simple systems (with polynomial $ϕ$'s) which were hypoelliptic but not subelliptic in the standard $L^2$-sense. So it is natural to analyze this problem of subellipticity which is in some sense intermediate (at least when $ϕ$ is $C^\infty$) between the maximal hypoellipticity (which was analyzed by Helffer-Nourrigat and Nourrigat) and the simple local hypoellipticity (or local microhypoellipticity) and to start first with the easiest non trivial examples. The analysis presented here is a continuation of a previous work by the first author and is devoted to the case of quasihomogeneous functions.

math.AP↗
arXiv2006

Singular Sums of Squares of Degenerate Vector Fields

We simplify and give an alternative proof of hypoellipticity for generalizations of the singular sum of squares of complex vector fields studied by Kohn, with an appendix by Derridj and Tartakoff, in the Annals of Mathematics, vol. 162 no. 2, 2005, pp. 943-986. The main generalization is that the complex vector fields now come from domains of finite type. We also prove real analytic hypoellipticity and the optimality of our estimates.

math.AP↗
arXiv2005

Analyticity and loss of derivatives

We prove local real analytic hypoellipticity for a sum of squares of complex vector fields studied by J.J. Kohn in a paper to appear in the Annals of Mathematics entitled "Hypoellipticity and loss of derivatives". The operator exhibits a loss of many derivatives but is nonetheless hypoelliptic, and, using L2 methods, we prove analytic hypoellipticity.

math.AP↗
arXiv2005

Analytic Hypoellipticity for a Class of Sums of Squares of Vector Fields with Non-Symplectic Characteristic Variety

The recent example of Hanges: $P = \partial_t^2 + t^2Δ_x + \partial^2_{θ(x)}$ in $R^3$ is analytic hypoelliptic in the sense of germs but not in the strong sense in any neighborhood of the origin. And its characteristic variety is non-symplectic. We give a purely $L^2,$ and hence quite flexible, proof of this result and generalizations, and link it to, and contrast it with, the celebrated Baouendi-Goulaouic operator. We point out that the results are consistent with the conjecture of Treves.

math.AP↗