We consider a second order operator with analytic coefficients whose principal symbol vanishes exactly to order two on a symplectic real analytic manifold. We assume that the first (non degenerate) eigenvalue vanishes on a symplectic submanifold of the characteristic manifold. In the $C^\infty$ framework this situation would mean a loss of 3/2 derivatives. We prove that this operator is analytic hypoelliptic. The main tool is the FBI transform. A case in which $C^\infty$ hypoellipticity fails is also discussed.
The global and semi-global analytic hypoellipticity on the torus is proved for two classes of sums of squares operators, introduced in "Analytic Hypoellipticity for Sums of Squares and the Treves Conjecture" by P. Albano and A. Bove and M. Mughetti, and in "Analytic Hypoellipticity for Sums of Squares and the Treves Conjecture. II" by A. Bove and M. Mughetti, satisfying the Hörmander condition and which fail to be neither locally nor microlocally analytic hypoelliptic.
On $T \times G$, where $T$ is a compact real-analytic manifold and $G$ is a compact Lie group, we consider differential operators $P$ which are invariant by left translations on $G$ and are elliptic in $T$. Under a mild technical condition, we prove that global hypoellipticity of $P$ implies its global analytic-hypoellipticity (actually Gevrey of any order $s \geq 1$). We also study the connection between the latter property and the notion of global analytic (resp. Gevrey) solvability, but in a much more general setup.
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…
In this paper, we present necessary and sufficient conditions to have global analytic hypoellipticity for a class of first-order operators defined on $\mathbb{T}^1 \times \mathbb{S}^3$. In the case of real-valued coefficients, we prove that an operator in this class is conjugated to a constant-coefficient operator satisfying a Diophantine condition, and that such conjugation preserves the global analytic hypoellipticity. In the case where the imaginary part of the coefficients is non-zero, we show that the operator is globally analytic hypoelliptic if the Nirenberg-Treves condition ($\mathcal{…
For each value of k, two complex vector fields satisfying the bracket condition are exhibited the sum of whose squares is hypoelliptic but not subelliptic - in fact the operator loses k-1 derivatives in Sobolev norms. In the Appendix it is proven that this operator is analytic hypoelliptic.
We consider an operator $ P $ which is a sum of squares of vector fields with analytic coefficients. The operator has a non-symplectic characteristic manifold, but the rank of the symplectic form $ σ$ is not constant on $ \Char P $. Moreover the Hamilton foliation of the non symplectic stratum of the Poisson-Treves stratification for $ P $ consists of closed curves in a ring-shaped open set around the origin. We prove that then $ P $ is analytic hypoelliptic on that open set. And we note explicitly that the local Gevrey hypoellipticity for $ P $ is $ G^{k+1} $ and that this is sharp.
We give a semiclassical analysis of a nonlinear eigenvalue problem arising from the study of the failure of analytic hypoellipticity and obtain a general family of hypoelliptic, but not analytic hypoelliptic operators.
We consider Keldysh-type operators, $ P = x_1 D_{x_1}^2 + a (x) D_{x_1} + Q (x, D_{x'} ) $, $ x = ( x_1, x') $ with analytic coefficients, and with $ Q ( x, D_{x'} ) $ second order, principally real and elliptic in $ D_{x'} $ for $ x $ near zero. We show that if $ P u =f $, $ u \in C^\infty $, and $ f $ is analytic in a neighbourhood of $ 0 $ then $ u $ is analytic in a neighbourhood of $ 0 $. This is a consequence of a microlocal result valid for operators of any order with Lagrangian radial sets. Our result proves a generalized version of a conjecture made by the second author and Lebeau and…
We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata in the Poisson-Treves stratification are symplectic. We produce a model operator, $ P_{1} $, having a single symplectic stratum and prove that it is Gevrey $ s_{0} $ hypoelliptic and not better. See Theorem \ref{th:1} for a definition of $ s_{0} $. We also show that this phenomenon has a microlocal character. We point out explicitly that this is a counterex…
In an interesting note, E.M. Stein observed some 20 years ago that while the Kohn Laplacian $\square_b$ on functions is neither locally solvable nor (analytic) hypoelliptic, the addition of a non-zero complex constant reversed these conclusions at least on the Heisenberg group, and Kwon reproved and generalized this result using the method of concatenations. Recently Hanges and Cordaro have studied this situation on the Heisenberg group in detail. Here we give a purely $L^2$ proof of Stein's result using the author's now classical construction of $(T^p)_ϕ= ϕT^p +...,$ where $T$ is the 'missing…
We attach representations to non-symplectic stratum in the characteristic set of real vector fields. This leads to Schrodinger operators. The analysis of the solutions of these Schrodinger equations allows us to construct smooth, non-analytic solutions of the sub-elliptic operator formed by taking sums of squares of the real vector fields.
Motivated by the problems of analytic hypoellipticity, we show that a special family of compact non self-adjoint operators has a non-zero eigenvalue. We recover old results by Christ,Hanges, Himonas, Pham-The-Lai and Robert proved by using ordinary differential equations. We show our method applies to higher dimensional cases, giving in particular a new class of hypoelliptic but not analytic hypoelliptic operators.
We investigate microlocal properties of partial differential operators with generalized functions as coefficients. The main result is an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth hypoelliptic symbols. Methodological novelties and technical refinements appear embedded into classical strategies of proof in order to cope with most delicate interferences by non-smooth lower order terms. We include simplified conditions which are applicable in special cases of interest.
A global real analytic regularity theorem for a quasilinear sum of squares of vector fields of Hormander rank 2 is given. A related local result for a special case was proved recently by the second author and L. Zanghirati in a paper titled "Local Real Analyticity of Solutions for sums of squares of non-linear vector fields".
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.
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.
This article studies hypoellipticity on general filtered manifolds. We extend the Rockland criterion to a pseudodifferential calculus on filtered manifolds, construct a parametrix and describe its precise analytic structure. We use this result to study Rockland sequences, a notion generalizing elliptic sequences to filtered manifolds. The main application that we present is to the analysis of the Bernstein--Gelfand--Gelfand (BGG) sequences over regular parabolic geometries. We do this by generalizing the BGG machinery to more general filtered manifolds (in a non-canonical way) and show that th…
In this note we present a symbolic pseudo-differential calculus on graded nilpotent Lie groups and, as an application, a version of the sharp Garding inequality. As a corollary, we obtain lower bounds for positive Rockland operators with variable coefficients as well as their Schwartz-hypoellipticity.
This article is concerned with quantitative unique continuation estimates for equations involving a "sum of squares" operator $\mathcal{L}$ on a compact manifold $\mathcal{M}$ assuming: $(i)$ the Chow-Rashevski-Hörmander condition ensuring the hypoellipticity of $\mathcal{L}$, and $(ii)$ the analyticity of $\mathcal{M}$ and the coefficients of $\mathcal{L}$. The first result is the tunneling estimate $\|φ\|_{L^2(ω)} \geq Ce^{- λ^{\frac{k}{2}}}$ for normalized eigenfunctions $φ$ of $\mathcal{L}$ from a nonempty open set $ω\subset \mathcal{M}$, where $k$ is the hypoellipticity index of $\mathcal…
The aim of this paper is to present the construction, out of the Kohn-Rossi complex, of a new hypoelliptic operator $Q_{L}$ on almost CR manifolds equipped with a real structure. The operator acts on all (p,q)-forms, but when restricted to (p,0)-forms and (p,n)-forms it is a sum of squares up to sign factor and lower order terms. Therefore, only a finite type condition condition is needed to have hypoellipticity on those forms. However, outside these forms $Q_{L}$ may fail to be hypoelliptic, as it is shown in the example of the Heisenberg group. We also look at the Fredholm properties of $Q_{…
This paper is devoted to the spatial analyticity of the solution of the BBM equation on the real line with an analytic initial data. It is shown that the analytic radius has a lower bound like $t^{-\frac{2}{3}}$ as time $t$ goes to infinity, which is an improvement of previous results. The main new ingredient is a higher order almost conservation law in analytic spaces. This is proved by introducing an equivalent analytic norm with smooth symbol and establishing some algebra identities of higher order polynomials.
We show that the uniform radius of spatial analyticity $σ(t)$ of solution at time $t$ for the fifth order KdV-BBM equation cannot decay faster than $1/t$ for large $t>0$, given initial data that is analytic with fixed radius $σ_0$. This significantly improves a recent result by Carvajal and Panthee, where they established an exponential decay of $σ(t)$ for large $t$.
We show that the uniform radius of spatial analyticity $σ(t)$ of solutions at time $t$ to the fifth order KdV-BBM equation cannot decay faster than $1/ \sqrt{t}$ for large $t$, given initial data that is analytic with fixed radius $σ_0$. This improves a recent result by Belayneh, Tegegn and the third author, where they obtained a $1/t$ decay of $σ(t)$ for large time $t$.
The point of this paper is to improve the reverse Agmon estimate discussed in \cite{TW} with assuming that the Schrodinger operator $P(h) = - h^2 Δ_g + V - E(h)$, $E(h)\to E$ as $h\to 0^+$, is analytic on a compact, real-analytic Riemannian manifold $(M,g)$. In this paper, by considering a Neumann problem with applying Poisson representation and exterior mass estimates on hypersurfaces, we can prove an improved reverse Agmon estimate on a hypersurface.
In this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time-dependent coefficients and lower order terms. We prove the Gevrey well-posedness of the Cauchy problem under $C^k$-regularity of coefficients of the principal part and natural Levi conditions on lower order terms which may be only continuous. In the case of analytic coefficients in the principal part we establish the $C^\infty$ well-posedness. The proofs are based on using the quasi-symmetriser for the corresponding system. The main novelty compared to the existing literature is the possibi…
It is shown that the uniform radius of spatial analyticity $σ(t)$ of solutions at time $t$ to the KdV equation cannot decay faster than $|t|^{-4/3}$ as $|t| \to \infty$ given initial data that is analytic with fixed radius $σ_0$. This improves a recent result of Selberg and Da Silva, where they proved a decay rate of $|t|^{-(4/3 + \varepsilon) }$ for arbitrarily small positive $\varepsilon$. The main ingredients in the proof are almost conservation law for the solution to the KdV equation in space of analytic functions and space-time dyadic bilinear $L^2$ estimates associated with the KdV equa…
In this paper, we study quantitative spatial analytic bounds and unique continuation inequalities of solutions for fractional heat equations with an analytic lower order term on the whole space. At first, we show that the solution has a uniform positive analytic radius for all time, and the solution enjoys a log-type ultra-analytic bound if the coefficient is ultra-analytic. Second, we prove a Hölder type interpolation inequality on a thick set, with an explicit dependence on the analytic radius of coefficient. Finally, by the telescoping series method, we establish an observability inequality…
The Triple Deck model is a classical high order boundary layer model that has been proposed to describe flow regimes where the Prandtl theory is expected to fail. At first sight the model appears to lose two derivatives through the pressure-displacement relation which links pressure to the tangential slip. In order to overcome this, we split the Triple Deck system into two coupled equations: a Prandtl type system on $\mathbb{H}$ and a Benjamin-Ono type equation on $\mathbb{R}$. This splitting enables us to extract a crucial leading order cancellation at the top of the lower deck. We develop a …
Explicit representations of densities for linear parabolic partial differential equations are useful in order to design computation schemes of high accuracy for a considerable class of diffusion models. Approximations of lower order based on the WKB-expansion have been used in order to compute Greeks in standard models of the interest rate market (cf. [2]). However, it turns out that for higher order approximations another related expansion leads to more accurate schemes. We compute a local explicit formula for a class of parabolic problems and determine a lower bound of the time horizon where…
Let $(Ω,g)$ be a compact, analytic Riemannian manifold with analytic boundary $\partial Ω= M.$ We give $L^2$-lower bounds for Steklov eigenfunctions and their restrictions to interior hypersurfaces $H \subset Ω^{\circ}$ in a geometrically defined neighborhood of $M$. Our results are optimal in the entire geometric neighborhood and complement the results on eigenfunction upper bounds in the author's previous work.
We consider the Euler equations on $\mathbb{T}^d$ with analytic data and prove lower bounds for the radius of spatial analyticity $ε(t)$ of the solution using a new method based on inductive estimates in standard Sobolev spaces. Our results are consistent with similar previous results proved by Kukavica and Vicol, but give a more precise dependence of $ε(t)$ on the radius of analyticity of the initial datum.
