This memoir deals with the hypoelliptic calculus on Heisenberg manifolds, or Heisenberg calculus. The Heisenberg manifolds generalize CR and contact manifolds and in this context the main differential operators at stake include the Hörmander's sum of squares, the Kohn Laplacian, the horizontal sublaplacian and its conformal powers and the contact Laplacian. These operators cannot be elliptic and the relevant pseudodifferential calculus to study them is provided by the Heisenberg calculus. The aim of this monograph is threefold. First, we give an intrinsic approach to the Heisenberg calculus by finding an intrinsic notion of principal symbol in this setting. This framework allows us to prove that the pointwise invertibility of a principal symbol, which can be restated in terms of the so-called Rockland condition, actually implies its global invertibility. Second, we study complex powers of hypoelliptic operators on Heisenberg manifolds in terms of the Heisenberg calculus. In particular, we show that complex powers of such operators give rise to holomorphic families in the Heisenberg calculus and we construct a scale of weighted Sobolev spaces providing us with sharp estimates for the operators in the Heisenberg calculus. Third, we make use of the Heisenberg calculus and of the results of this monograph to derive spectral asymptotics for hypoelliptic operators on Heisenberg manifolds. The advantage of using the Heisenberg calculus is illustrated by reformulating in a geometric fashion these asymptotics for the main geometric operators on CR and contact manifolds.
This paper is part of a series papers devoted to geometric and spectral theoretic applications of the hypoelliptic calculus on Heisenberg manifolds. More specifically, in this paper we make use of the Heisenberg calculus of Beals-Greiner and Taylor to analyze the spectral theory of hypoelliptic operators on Heisenberg manifolds. The main results of this paper include: (i) Obtaining complex powers of hypoelliptic operators as holomorphic families of Psi_{H}DO's, which can be used to define a scale of weighted Sobolev spaces interpolating the weighted Sobolev spaces of Folland-Stein and providin…
We establish the existence of a bounded $H_\infty$-calculus for a large class of hypoelliptic pseudodifferential operators on R^n and closed manifolds.
In this paper, we show the nonexistence results for the Kirchhoff elliptic, parabolic, and hyperbolic type equations on the Heisenberg groups. Also, the pseudo-parabolic and pseudo-hyperbolic equations of the Kirchhoff-type are under consideration. To prove these results we use the test function method. In addition, the analogous results are transferred to the cases of systems. Also, we give some examples of non-local nonlinearities.
We extend the classical regularity theorem of elliptic operators to maximally hypoelliptic differential operators. More precisely, given vector fields $X_1,\ldots,X_m$ on a smooth manifold which satisfy Hörmander's bracket generating condition, we define a principal symbol for \textit{any} linear differential operator. Our symbol takes into account the vector fields $X_i$ and their commutators. We show that for an arbitrary differential operator, its principal symbol is invertible if and only if the operator is maximally hypoelliptic. This answers affirmatively a conjecture due to Helffer and …
The aim of this work is to develop a global calculus for pseudo-differential operators acting on suitable algebras of generalized functions. In particular, a condition of global hypoellipticity of the symbols gives a result of regularity for the corresponding pseudo-differential equations. This calculus and this frame are proposed as tools for the study in Colombeau algebras of partial differential equations globally defined on $\mathbb{R}^n$.
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 discusses the existence of gradient estimates for second order hypoelliptic heat kernels on manifolds. It is now standard that such inequalities, in the elliptic case, are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associated "Ricci curvature" takes on the value -\infty at points of degeneracy of the semi-Riemannian metric associated to the operator. For this reason, the standard proofs for the elliptic theory fail in the hypoelliptic setting. This paper presents recent results for hypoelliptic operators. Malliavin calcu…
In this paper we study nonlocal equations driven by the fractional powers of hypoelliptic operators in the form $$\mathscr K u = \mathscr A u - \partial_t u \overset{def}{=} \operatorname{tr}(Q \nabla^2 u) + <BX,\nabla u> - \partial_t u,$$ introduced by Hörmander in his 1967 hypoellipticity paper. We show that the nonlocal operators $(-\mathscr K)^s$ and $(-\mathscr A)^s$ can be realized as the Dirichlet-to-Neumann map of doubly-degenerate extension problems. We solve such problems in $L^\infty$, and in $L^p$ for $1\leq p<\infty$ when $\operatorname{tr}(B)\geq 0$. In forthcoming works we use s…
In his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order evolution equation which displays some challenging features. In the opening of his 1967 hypoellipticity paper Hörmander discussed a general class of degenerate Ornstein-Uhlenbeck operators that includes Kolmogorov's as a special case. In this note we combine semigroup theory with a nonlocal calculus for these hypoelliptic operators to establish new inequalities of Hardy-Littlewood-Sobolev type in the situation when the drift matrix has nonnegative trace. Our work has been influenced by ide…
In this paper we prove a short time asymptotic expansion of a hypoelliptic heat kernel on an Euclidean space and a compact manifold. We study the "cut locus" case, namely, the case where energy-minimizing paths which join the two points under consideration form not a finite set, but a compact manifold. Under mild assumptions we obtain an asymptotic expansion of the heat kernel up to any order. Our approach is probabilistic and the heat kernel is regarded as the density of the law of a hypoelliptic diffusion process, which is realized as a unique solution of the corresponding stochastic differe…
In this paper we define an intrinsic notion of principal for the Hypoelliptic calculus on Heisenberg manifolds. More precisely, the principal symbol of a \psivdo appears as a homogeneous section over the linear dual of the tangent Lie algebra bundle of the manifold. This definition is an important step towards using global $K$-theoretic tools in the Heisenberg setting, such as those involved in the elliptic setting for proving the Atiyah-Singer index theorem or the regularity of the eta invariant. On the other hand, the intrinsic definition of the principal symbol enables us to give an intrins…
In this work we obtain continuity results on Hölder spaces for operators belonging to a Weyl-Hörmander calculus for metrics such that the class of the associated operators contains, in particular, certain hypoelliptic laplacians. With our results we recover some historical Hölder boundedness theorems (see R. Beals \cite{Be,Be2}). The action of (periodic) Ruzhansky-Turunen classes of pseudo-differential operators on Hölder spaces also will be investigated.
We explore the geometry that underlies the osculating nilpotent group structures of the Heisenberg calculus. For a smooth manifold $M$ with a distribution $H\subseteq TM$ analysts use explicit (and rather complicated) coordinate formulas to define the nilpotent groups that are central to the calculus. Our aim in this paper is to provide insight in the intrinsic geometry that underlies these coordinate formulas. First, we introduce `parabolic arrows' as a generalization of tangent vectors. The definition of parabolic arrows involves a mix of first and second order derivatives. Parabolic arrows …
The aim of this paper is to offer an original and comprehensive spectral theoretical approach to the study of convergence to equilibrium, and in particular of the hypocoercivity phenomenon, for contraction semigroups in Hilbert spaces. Our approach rests on a commutation relationship for linear operators known as intertwining, and we utilize this identity to transfer spectral information from a known, reference semigroup $\tilde{P} = (e^{-t\tilde{\mathbf A}})_{t \geq 0}$ to a target semigroup $P$ which is the object of study. This allows us to obtain conditions under which $P$ satisfies a hypo…
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…
In this note we present a symbolic pseudo-differential calculus on the Heisenberg group. We particularise to this group our general construction [4,3,2] of pseudo-differential calculi on graded groups. The relation between the Weyl quantization and the representations of the Heisenberg group enables us to consider here scalar-valued symbols. We find that the conditions defining the symbol classes are similar but different to the ones in [1]. Applications are given to Schwartz hypoellipticity and to subelliptic estimates on the Heisenberg group.
In this paper we study the Cauchy problem for the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups when the time-dependent non-negative propagation speed is regular, Hölder, and distributional. For Hölder coefficients we derive the well-posedness in the spaces of ultradistributions associated to Rockland operators on graded groups. In the case when the propagation speed is a distribution, we employ the notion of "very weak solutions" to the Cauchy problem, that was already successfully used in similar contexts in [GR15] and [RT17b]. We show that the Cau…
Polyhomogeneous symbols, defined by Kohn-Nirenberg and Hörmander in the 60's, play a central role in the symbolic calculus of most pseudodifferential calculi. We prove a simple characterisation of polyhomogeneous functions which avoids the use of asymptotic expansions. Specifically, if $U$ is open subset of $\mathbb{R}^d$, then a polyhomogeneous symbol on $U \times \mathbb{R}^d$ is precisely the restriction to $t=1$ of a function on $U \times \mathbb{R}^{d+1}$ which is homogeneous for the dilations of $\mathbb{R}^{d+1}$ modulo Schwartz class functions. This result holds for arbitrary graded di…
In this paper we show the Hörmander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking, under general Hörmander's Lie bracket conditions, we show the regularization effect of discontinuous Lévy noises for possibly degenerate stochastic differential equations with jumps. To treat the large jumps, we use the perturbation argument together with interpolation techniques and some short time asymptotic estimates of the semigroup. As an application, we show the existence of fundamental solutions for operator $\partial_t-\mathscr{K}$,…
A class of inverse problems for restoring the right-hand side of a parabolic equation for a large class of positive operators with discrete spectrum is considered. The results on existence and uniqueness of solutions of these problems as well as on the fractional time diffusion (subdiffusion) equations are presented. Consequently, the obtained results are applied for the similar inverse problems for a large class of subelliptic diffusion and subdiffusion equations (with continuous spectrum). Such problems are modelled by using general homogeneous left-invariant hypoelliptic operators on genera…
By further developing the generalized $Γ$-calculus for hypoelliptic operators, we prove hypocoercive estimates for a large class of Kolmogorov type operators which are defined on non necessarily totally geodesic Riemannian foliations. We study then in detail the example of the velocity spherical Brownian motion, whose generator is a step-3 generating hypoelliptic Hörmander's type operator. To prove hypocoercivity in that case, the key point is to show the existence of a convenient Riemannian foliation associated to the diffusion. We will then deduce, under suitable geometric conditions, the co…
The theme of this work is that the theory of charged particles in a uniform magnetic field can be generalized to a large class of operators if one uses an extended a class of Weyl operators which we call "Landau--Weyl pseudodifferential operators". The link between standard Weyl calculus and Landau--Weyl calculus is made explicit by the use of an infinite family of intertwining "windowed wavepacket transforms"; this makes possible the use of the theory of modulation spaces to study various regularity properties. Our techniques allow us not only to recover easily the eigenvalues and eigenfuncti…
A class of pseudodifferential operators on the Heisenberg group is defined. As it should be, this class is an algebra containing the class of differential operators. Furthermore, those pseudodifferential operators act continuously on Sobolev spaces and the loss of derivatives may be controled by the order of the operator. Although a large number of works have been devoted in the past to the construction and the study of algebras of variable-coefficient operators, including some very interesting works on the Heisenberg group, our approach is different, and in particular puts into light microloc…
This paper has been withdrawn by the authors. A class of pseudodifferential operators on the Heisenberg group is defined. As it should be, this class is an algebra containing the class of differential operators. Furthermore, those pseudodifferential operators act continuously on Sobolev spaces and the loss of derivatives may be controled by the order of the operator. Although a large number of works have been devoted in the past to the construction and the study of algebras of variable-coefficient operators, including some very interesting works on the Heisenberg group, our approach is differe…
The overall goal of this dissertation is to investigate certain classical results from harmonic analysis, replacing the Euclidean setting, the abelian structure and the elliptic Laplace operator with a non-commutative environment and hypoelliptic operators. More specifically, we consider wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent non-negative propagation speeds that are Hölder-regular or even more so. The corresponding Euclidean problem has been extensively studied in the `80s and some additional results have been recently obta…
We outline an abstract approach to the pseudo-differential Weyl calculus for operators in function spaces in infinitely many variables. Our earlier approach to the Weyl calculus for Lie group representations is extended to the case of representations associated with infinite-dimensional coadjoint orbits. We illustrate the approach by the case of infinite-dimensional Heisenberg groups. The classical Weyl-Hörmander calculus is recovered for the Schrödinger representations of the finite-dimensional Heisenberg groups.
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.
The short-time heat kernel expansion of elliptic operators provides a link between local and global features of classical geometries. For many geometric structures related to (non-)involutive distributions, the natural differential operators tend to be Rockland, hence hypoelliptic. In this paper we establish a universal heat kernel expansion for formally selfadjoint non-negative Rockland differential operators on general closed filtered manifolds. The main ingredient is the analysis of parametrices in a recently constructed calculus adapted to these geometric structures. The heat expansion imp…
The purpose of this note is to compare the properties of the symbolic pseudo-differential calculus on the Heisenberg and on the Engel groups; nilpotent Lie groups of 2-step and 3-step, respectively. Here we provide a preliminary analysis of the structure and of the symbolic calculus with symbols parametrized by $(λ,μ)$ on the Engel group, while for the case of the Heisenberg group we recall the analogous results on the $λ$-classes of symbols.
In this note we point out and fill a gap in the proof by Julg-Kasparov of the Baum-Connes conjecture with coefficients for discrete subgroups of $\op{SU}(n,1)$. The issue at stake is the proof that the complex powers of the contact Laplacian are element of the Heisenberg calculus. In particular, we explain why we cannot implement into the setting of the Heisenberg calculus the classical Seeley's approach to complex powers.
We describe boundedness and compactness properties for the operators obtained by the Weyl-Pedersen calculus in the case of the irreducible unitary representations of nilpotent Lie groups that are associated with flat coadjoint orbits. We use spaces of smooth symbols satisfying appropriate growth conditions expressed in terms of invariant differential operators on the coadjoint orbit under consideration. Our method also provides conditions for these operators to belong to one of the Schatten ideals of compact operators. In the special case of the Schrödinger representation of the Heisenberg gro…
We produce, on general homogeneous groups, an analogue of the usual Hörmander pseudodifferential calculus on Euclidean space, at least as far as products and adjoints are concerned. In contrast to earlier works, we do not limit ourselves to analogues of classical symbols, nor to the Heisenberg group. The key technique is to understand ``multipliers'' of any given order j, and the operators of convolution with their inverse Fourier transforms, which we here call convolution operators of order j. (Here a ``multiplier'' is an analogue of a Hörmander-type symbol a(x,ξ), which is independent of x.)…
We study scattering and inverse scattering theories for asymptotically complex hyperbolic manifolds. We show the existence of the scattering operator as a meromorphic family of operators in the Heisenberg calculus on the boundary, which is a contact manifold with a pseudohermitian structure. Then we define the radiation fields as in the real asymptotically hyperbolic case, and reconstruct the scattering operator from those fields. As an application we show that the manifold, including its topology and the metric, are determined up to invariants by the scattering matrix at all energies.
This paper is an extended version of math.OA/0601528 where we point out and remedy a gap in the proof by P. Julg and G. Kasparov of the Baum-Connes conjecture for discrete subgroups of SU(n,1). In particular, here we explain in details why the non-microlocality of the Heisenberg calculus prevents us from implementing into this framework the classical approach of Seeley to pseudodifferential complex powers, which was the main issue at stake in math.OA/0601528.
In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form $$ Au(x)=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}e^{i(x-y)\cdotξ}σ(x+τ(y-x),ξ)u(y)dydξ, $$ where $τ:\mathbb{R}^n\to\mathbb{R}^n$ is a general function. In particular, for the linear choices $τ(x)=0$, $τ(x)=x$, and $τ(x)=\frac{x}{2}$ this covers the well-known Kohn-Nirenberg, anti-Kohn-Nirenberg, and Weyl quantizations, respectively. Quantizations of such type appear naturally in the analysis on nilpotent Lie groups for polynomial functions $τ$ and here we investigate the correspo…
