Gevrey Vectors Research Articles

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 Derridj (Pac J Math 302(2):511–543, 2019) 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.

A FOURIER-TYPE CHARACTERISATION FOR GEVREY VECTORS ON HYPO-ANALYTIC STRUCTURES AND PROPAGATION OF GEVREY SINGULARITIES

AbstractIn this work we prove a Fourier–Bros–Iagolnitzer (F.B.I.) characterisation for Gevrey vectors on hypo-analytic structures and we analyse the main differences of Gevrey regularity and hypo-analyticity concerning the F.B.I. transform. We end with an application of this characterisation on a propagation of Gevrey singularities result for solutions of the nonhomogeneous system associated with the hypo-analytic structure for analytic structures of tube type.

Gevrey regularity of Gevrey vectors of second-order partial differential operators with non-negative characteristic form

Given a second-order partial differential operator $$P = P^\mathrm{o} + X + a$$ on an open set $$\Omega $$ in $${\mathbb {R}}^n$$ , where $$P^\mathrm{o}$$ is the principal part and X is a real vector field, with non-negative real characteristic form, we study the $$s'$$ -Gevrey regularity on an open subset $$\omega $$ of $$\Omega $$ , of s-Gevrey vectors of P on $$\omega $$ . For that we associate to any subset $$A \subset \Omega $$ an integer (finite or $$+\infty $$ ) named the type of A with respect to $$P^\mathrm{o}$$ and denoted $$\tau (A; P^\mathrm{o}$$ ) (see in the next sections, precise definitions, facts and remarks about it). Denoting the space of s-Gevrey vectors of P in $$\omega $$ by $$G^s(\omega ,P)$$ , we prove that $$G^{s}(\omega ; P) \subset G^{s'} (\omega )$$ , with $$s' = \tau (\omega ; P)\cdot s$$ under the assumption that the coefficients of P are in $$G^{s} (\omega )$$ . Moreover, $$s'$$ is optimal.

Local estimates for Hörmander’s operators of first kind with analytic Gevrey coefficients and application to the regularity of their Gevrey vectors

Local estimates for Hörmander’s operators of first kind with analytic Gevrey coefficients and application to the regularity of their Gevrey vectors

On local Gevrey regularity for Gevrey vectors of subelliptic sums of squares: an elementary proof of a sharp Gevrey Kotake–Narasimhan theorem

We study the regularity of Gevrey vectors for Hormander operators $$\begin{aligned} P = \sum _{j=1}^m X_j^2 + X_0 + c \end{aligned}$$ where the $$X_j$$ are real vector fields and c(x) is a smooth function, all in Gevrey class $$G^{s}.$$ The principal hypothesis is that P satisfies the subelliptic estimate: for some $$\varepsilon >0, \; \exists \,C$$ such that $$\begin{aligned} \Vert v\Vert _{\varepsilon }^2 \le C\left( |(Pv, v)| + \Vert v\Vert _0^2\right) \qquad \forall v\in C_0^\infty . \end{aligned}$$ We prove directly (without the now familiar use of adding a variable t and proving suitable hypoellipticity for $$Q=-D_t^2-P$$ and then, using the hypothesis on the iterates of P on u, constructing a homogeneous solution U for Q whose trace on $$t=0$$ is just u) that for $$s\ge 1,$$ $$G^s(P,\Omega _0) \subset G^{s/\varepsilon }(\Omega _0);$$ that is, $$\begin{aligned}&\forall K\Subset \Omega _0, \;\exists C_K: \Vert P^j u\Vert _{L^2(K)}\le C_K^{j+1} (2j)!^s, \;\forall j\\&\quad \implies \forall K'\Subset \Omega _0, \;\exists \tilde{C}_{K'}:\,\Vert D^\ell u\Vert _{L^2(K')} \le \tilde{C}_{K'}^{\ell +1} \ell !^{s/\varepsilon }, \;\forall \ell . \end{aligned}$$ In other words, Gevrey growth of derivatives of u as measured by iterates of P yields Gevrey regularity for u in a larger Gevrey class. When $$\varepsilon =1,$$ P is elliptic and so we recover the original Kotake–Narasimhan theorem (Kotake and Narasimhan in Bull Soc Math Fr 90(12):449–471, 1962), which has been studied in many other classes, including ultradifferentiable functions (Boiti and Journet in J Pseudo-Differ Oper Appl 8(2):297–317, 2017). We are indebted to M. Derridj for multiple conversations over the years.

Gevrey vectors in involutive tube structures and Gevrey regularity for the solutions of certain classes of semilinear systems

In this work, we introduce the notion of s-Gevrey vectors in locally integrable structures of tube type. Under the hypothesis of analytic hypoellipticity, we study the Gevrey regularity of such vectors and also show how this notion can be applied to the study of the Gevrey regularity of solutions of certain classes of semilinear equations.

Gevrey vectors of multi-quasi-elliptic systems

We show that the multi-quasi-ellipticity is a necessary and sufficient condition for the property of elliptic iterates to hold for multi-quasi-homogenous differential operators.

Une g�n�ralisation du probl�me des it�r�s

For given systems of differential operators of constant coefficients we give algebraic conditions to compare their spaces of Gevrey vectors.

Weighted Strongly Elliptic Operators on Lie Groups

Let (H, G, U) be a continuous representation of a Lie group G by bounded operators g ↦ U(g) on the Banach space X and let (X, g, dU) denote the representation of the Lie algebra g obtained by differentiation. If a1, …, ad′ is a Lie algebra basis of g, Ai = dU(ai) and Aα = Ai1 …Aik whenever α = (i1, …, ik) we consider the operators [formula] where the cα are complex coefficients satisfying a weighted strongly elliptic condition in which different directions may have different weights. This condition is such that the class of operators considered encompasses all the standard strongly elliptic operators. We prove that the closure H̄ of each such operator H generates a holomorphic semigroup S with holomorphy sector which contains a non-empty subsector determined by the coefficients and independent of the representation. Moreover, the semigroup S has a smooth representation independent kernel and we derive bounds on the kernel and all its derivatives. Finally we establish elliptic regularity properties for the operators and their powers and characterize the analytic and Gevrey vectors. As a corollary we derive optimal growth bounds for the eigenfunctions of the anharmonic oscillators P2m + Q2n.

On Infinitely Differentiable and Gevrey Vectors for Representation

On Infinitely Differentiable and Gevrey Vectors for Representation

On infinitely differentiable and Gevrey vectors for representation

In the present paper we give a condition in order that the set of infinitely differentiable vectors for a representation π \pi in a Banach space be equal to the set of all infinitely differentiable vectors for the restriction of π \pi to a subgroup. Similar results for Gevrey vectors and analytic vectors are proved for unitary representations.