On the Gevrey regularity for Sums of Squares of vector fields, study of some models

The micro-local Gevrey regularity of a class of "sums of squares" with real analytic coefficients is studied in detail. Some partial regularity result is also given

Sulla regolarità delle soluzioni e dei vettori analitici per “somme di quadrati”

We present a brief survey on some recent results concerning the local and global regularity of the solutions for some classes/models of sums of squares of vector fields with real-valued real analytic coefficients of H"ormander type. Moreover we also illustrate 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 H"ormander type, thus providing a microlocal version, in the analytic category, of a result due to M. Derridj.Presentiamo una breve rassegna di alcuni recenti risultati riguardanti la regolarità locale e globale delle soluzioni per alcune classi/modelli di somme di quadrati di campi vettoriali con coefficienti reali analitici a valori reali di tipo H"ormander. Illustriamo anche un risultato riguardante la regolarità microlocale dei vettori analitici per operatori somme di quadrati di campi vettoriali con coefficienti reali analitici a valori reali di tipo H"ormander, fornendo così una versione microlocale, nel caso analitico, di un risultato dovuto a M. Derridj

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 $\Omega$ 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

Gevrey regularity for a generalization of the Oleĭnik–Radkevič operator

Abstract The Gevrey hypoellipticity of a class of models generalizing the Oleĭnik–Radkevic operator is studied. Some partial regularity result is also given. It is studied the partial and microlocal regularity of the operator L ( t , x ; D t , D x ) = D t 2 + ∑ j = 1 n t 2 ( r j − 1 ) D x j 2 on Ω , open neighborhood of the origin in R n + 1 , where the r j 's are positive integers such that r 1 r 2 ⋯ r n

Analytic and Gevrey Hypoellipticity for Perturbed Sums of Squares Operators

We prove a couple of results concerning pseudodifferential perturbations of differential operators being sums of squares of vector fields and satisfying H\"ormander's condition. The first is on the minimal Gevrey regularity: if a sum of squares with analytic coefficients is perturbed with a pseudodifferential operator of order strictly less than its subelliptic index it still has the Gevrey minimal regularity. We also prove a statement concerning real analytic hypoellipticity for the same type of pseudodifferential perturbations, provided the operator satisfies to some extra conditions (see Theorem 1.2 below) that ensure the analytic hypoellipticity

Impact of Small Repeat Sequences on Bacterial Genome Evolution

Intergenic regions of prokaryotic genomes carry multiple copies of terminal inverted repeat (TIR) sequences, the nonautonomous miniature inverted-repeat transposable element (MITE). In addition, there are the repetitive extragenic palindromic (REP) sequences that fold into a small stem loop rich in G–C bonding. And the clustered regularly interspaced short palindromic repeats (CRISPRs) display similar small stem loops but are an integral part of a complex genetic element. Other classes of repeats such as the REP2 element do not have TIRs but show other signatures. With the current availability of a large number of whole-genome sequences, many new repeat elements have been discovered. These sequences display diverse properties. Some show an intimate linkage to integrons, and at least one encodes a small RNA. Many repeats are found fused with chromosomal open reading frames, and some are located within protein coding sequences. Small repeat units appear to work hand in hand with the transcriptional and/or post-transcriptional apparatus of the cell. Functionally, they are multifaceted, and this can range from the control of gene expression, the facilitation of host/pathogen interactions, or stimulation of the mammalian immune system. The CRISPR complex displays dramatic functions such as an acquired immune system that defends against invading viruses and plasmids. Evolutionarily, mobile repeat elements may have influenced a cycle of active versus inactive genes in ancestral organisms, and some repeats are concentrated in regions of the chromosome where there is significant genomic plasticity. Changes in the abundance of genomic repeats during the evolution of an organism may have resulted in a benefit to the cell or posed a disadvantage, and some present day species may reflect a purification process. The diverse structure, eclectic functions, and evolutionary aspects of repeat elements are described

Germ hypoellipticity and loss of derivatives

We prove hypoellipticity in the sense of germs for the operator $$P = L q L ¯ q + L ¯ q t 2 k L q + Q 2 , \mathcal {P}= L_{q}\overline {L}_{q} + \overline {L}_{q}t^{2k}L_{q} +Q^{2},$$ where $$L q = D t + i t q − 1 − Δ x and Q = x 1 D 2 − x 2 D 1 , L_{q}=D_{t}+it^{q-1}\sqrt {-\Delta _{x}}\quad \text {and}\quad Q = x_{1}D_{2}-x_{2}D_{1},$$ even though it fails to be hypoelliptic in the strong sense. The primary tool is an a priori estimate

On the sharp Gevrey regularity for a generalization of the M\'etivier operator

The sharp Gevrey hypoellipticity is provided for the following generalization of the M\'etivier operator, "Non-hypoellipticit\'e analytique pour $D_{x}^{2}+\left( x^{2} + y^{2}\right)D_{y}^{2}$" by G. M\'etivier, \begin{align*} D_{x}^{2}+\left(x^{2n+1}D_{y}\right)^{2}+\left(x^{n}y^{m}D_{y}\right)^{2}, \end{align*} in $\Omega$ open neighborhood of the origin in $\mathbb{R}^{2}$, where $n$ and $m$ are positive integers

Sulla regolarità delle soluzioni e dei vettori analitici per “somme di quadrati”

We present a brief survey on some recent results concerning the local and global regularity of the solutions for some classes/models of sums of squares of vector fields with real-valued real analytic coefficients of H"ormander type. Moreover we also illustrate 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 H"ormander type, thus providing a microlocal version, in the analytic category, of a result due to M. Derridj.Presentiamo una breve rassegna di alcuni recenti risultati riguardanti la regolarità locale e globale delle soluzioni per alcune classi/modelli di somme di quadrati di campi vettoriali con coefficienti reali analitici a valori reali di tipo H"ormander. Illustriamo anche un risultato riguardante la regolarità microlocale dei vettori analitici per operatori somme di quadrati di campi vettoriali con coefficienti reali analitici a valori reali di tipo H"ormander, fornendo così una versione microlocale, nel caso analitico, di un risultato dovuto a M. Derridj