We prove global subelliptic estimates for quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. In a previous joint work with M. Hitrik, we pointed out the existence of a particular linear subvector space in the phase space intrinsically associated to their Weyl symbols, called singular space, which rules spectral properties of non-elliptic quadratic operators. The purpose of the present paper is to prove that quadratic operators whose singular spaces are reduced to zero, are subelliptic with a loss of "derivatives" depending directly on particular algebraic properties of the Hamilton maps of their Weyl symbols. More generally, when singular spaces are symplectic spaces, we prove that quadratic operators are subelliptic in any direction of the symplectic orthogonal complements of their singular spaces.
Using an approach based on the techniques of FBI transforms, we give a new simple proof of the global subelliptic estimates for non-selfadjoint non-elliptic quadratic differential operators, under a natural averaging condition on the Weyl symbols of the operators, established by the second author. The loss of the derivatives in the subelliptic estimates depends directly on algebraic properties of the Hamilton maps of the quadratic symbols. Using the FBI point of view, we also give accurate smoothing estimates of Gelfand-Shilov type for the associated heat semigroup in the limit of small times.
In this article we study some Kramers-Fokker-Planck operators with a polynomial potential $V(q)$ of degree greater than two having quadratic limiting behavior. This work provides an accurate global subelliptic estimate for KFP operators under some conditions imposed on the potential.
We study semigroups generated by general fractional Ornstein-Uhlenbeck operators acting on $L2(\mathbb R^n)$. We characterize geometrically the partial Gevrey-type smoothing properties of these semigroups and we sharply describe the blow-up of the associated seminorms for short times, generalizing the hypoelliptic and the quadratic cases. As a byproduct of this study, we establish partial subelliptic estimates enjoyed by fractional Ornstein-Uhlenbeck operators on the whole space by using interpolation theory.
In this paper we study the Cauchy problem for the wave equations for sums of squares of left invariant vector fields on compact Lie groups and also for hypoelliptic homogeneous left-invariant differential operators on graded Lie groups (the positive Rockland operators), when the time-dependent propagation speed satisfies a Log-Lipschitz condition. We prove the well-posedness in the associated Sobolev spaces exhibiting a finite loss of regularity with respect to the initial data, which is not true when the propagation speed is a ${\rm H\ddot{o}lder}$ function. We also indicate an extension to g…
We characterize geometrically the regularizing effects of the semigroups generated by accretive non-selfadjoint quadratic differential operators. As a byproduct, we establish the subelliptic estimates enjoyed by these operators, being expected to be optimal. These results prove conjectures by M. Hitrik, K. Pravda-Starov and J. Viola. The proof relies on a new representation of the polar decomposition of these semigroups. In particular, we identify the selfadjoint part as the evolution operator generated by the Weyl quantization of a time-dependent real-valued nonnegative quadratic form for whi…
This is a chapter from PhD Thesis by Stefano Biagi (advisor: prof. A. Bonfiglioli). We overview existing results showing that it is possible to generalize the classical Hardy's Inequality to more general linear partial differential operators (PDOs, in the sequel), possibly degenerate-elliptic, of the following quasi-divergence form $$ \mathcal{L} = \frac{1}{w(x)}\sum_{i = 1}^N\frac{\partial}{\partial x_i} \left(\sum_{j = 1}^Nw(x)a_{ij}(x)\frac{\partial}{\partial x_j}\right), \quad x \in \mathbb{R}^N, $$ where $w \in C^{\infty}(\mathbb{R}^N,\mathbb{R})$ is a (smooth and) strictly positive funct…
In this paper the dependence of the best constants in Sobolev and Gagliardo-Nirenberg inequalities on the precise form of the Sobolev space norm is investigated. The analysis is carried out on general graded Lie groups, thus including the cases of $\mathbb R^n$, Heisenberg, and more general stratified Lie groups. The Sobolev norms may be defined in terms of Rockland operators, i.e. the hypoelliptic homogeneous left-invariant differential operators on the group. The best constants are expressed in the variational form as well as in terms of the ground state solutions of the corresponding nonlin…
We prove existence of solutions for a class of systems of subelliptic PDEs arising from Mean Field Game systems with Hörmander diffusion. These results are motivated by the feedback synthesis Mean Field Game solutions and the Nash equilibria of a large class of $N$-player differential games.
We study accretive quadratic operators with zero singular spaces. These degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in the Schwartz space for any positive time. In this work, we study the short-time asymptotics of the regularizing effect induced by these semigroups. We show that these short-time asymptotics of the regularizing effect depend on the directions of the phase space, and that this dependence can be nicely understood through the structure of the singular space. As a byproduct of these result…
We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relevant the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeters on stratified Lie groups.
In this work we investigate the well-posedness for difussion equations associated to subelliptic pseudo-differential operators on compact Lie groups. The diffusion by strongly elliptic operators is considered as a special case and in particular the fractional diffusion with respect to the Laplacian. The general case is studied within the Hörmander classes associated to a sub-Riemannian structure on the group (encoded by a Hörmander system of vector fields). Applications to diffusion equations for fractional sub-Laplacians, fractional powers of more general subelliptic operators, and the corres…
We prove a couple of results concerning pseudodifferential perturbations of differential operators being sums of squares of vector fields and satisfying Hörmander'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 Theo…
We study the partial Gelfand-Shilov regularizing effect and the exponential decay for the solutions to evolution equations associated to a class of accretive non-selfadjoint quadratic operators, which fail to be globally hypoelliptic on the whole phase space. By taking advantage of the associated Gevrey regularizing effects, we study the null-controllability of parabolic equations posed on the whole Euclidean space associated to this class of possibly non-globally hypoelliptic quadratic operators. We prove that these parabolic equations are null-controllable in any positive time from thick con…
Genus one polyhedral surfaces, spaces of quadratic differentials on tori and determinants of Laplacians
证明了亏格为1的任意紧致多面体曲面上拉普拉斯算子行列式的一个公式。这个公式推广了著名的Ray-Singer关于平坦环面的结果。还考虑了由椭圆曲面上亚纯二次微分的模给出的平坦圆锥度量的一种特殊情况。研究了具有L个单零点和L个单极点的亚纯二次微分的模空间上Laplacian算子作为泛函的行列式，并给出了该泛函的变分公式. R. t.该模空间上的自然坐标。给出了任意亏格的紧致Riemann曲面上关于平坦锥度量的Troyanov定理的一个新证明。
We introduce some general tools to design exact splitting methods to compute numerically semigroups generated by inhomogeneous quadratic differential operators. More precisely, we factorize these semigroups as products of semigroups that can be approximated efficiently, using, for example, pseudo-spectral methods. We highlight the efficiency of these new methods on the examples of the magnetic linear Schr{ö}dinger equations with quadratic potentials, some transport equations and some Fokker-Planck equations.
