Elliptic Pseudodifferential Operators Research Articles

Extremal metrics for the Q′-curvature in three dimensions

We construct contact forms with constant Q′-curvature on compact three-dimensional CR manifolds that admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the II-functional from conformal geometry. Two crucial steps are to show that the P′-operator can be regarded as an elliptic pseudodifferential operator and to compute the leading-order terms of the asymptotic expansion of the Green's function for P′.

The Poisson transform on a compact real analytic Riemannian manifold

We study the transform mapping an \(L^{2}\) function f on a compact, real analytic Riemannian manifold X to the analytic continuation of \(\exp (-t \sqrt{\Delta })f\) to the interior of a Grauert tube tube \(M_{t}\) about X. We show that after precomposing with an elliptic pseudodifferential operator this becomes a unitary map from \(L^{2}(X)\) onto the holomorphic \(L^{2}\) functions on \(M_{t}\). If a compact Lie group of isometries acts transitively on X then the inverse of this unitarized map can be constructed by the restrict ion principle.

Variations on Quantum Ergodic Theorems

We derive some quantum ergodic theorems, related to microlocal behavior of eigenfunctions of a positive, self-adjoint, elliptic pseudodifferential operator Λ on a compact Riemannian manifold M, emphasizing results that hold without the hypothesis that the Hamiltonian flow generated by the symbol of Λ be ergodic. Cases treated include both integrable Hamiltonians and some associated with “soft chaos.”

Regularity for the Supercritical Fractional Laplacian with Drift

We consider the linear stationary equation defined by the fractional Laplacian with drift. In the supercritical case, wherein the dominant term is given by the drift instead of the diffusion component, we prove local regularity of solutions in Sobolev spaces employing tools from the theory of pseudo-differential operators. The regularity of solutions in the supercritical case is as expected from the subcritical case. In the subcritical case the diffusion is at least as strong as the drift, and the operator is an elliptic pseudo-differential operator, which is not the case in the supercritical regime. We also compute the leading part of the singularity of the Green’s kernel for the supercritical case, which displays some rather unusual behavior.

Zeta functions, excision in cyclic cohomology and index problems

The aim of this paper is to show how zeta functions and excision in cyclic cohomology may be combined to obtain index theorems. In the first part, we obtain an index formula for “abstract elliptic pseudodifferential operators” associated to spectral triples, in the spirit of the one of Connes and Moscovici. This formula is notably well adapted when the zeta function has multiple poles. The second part is devoted to give a concrete realization of this formula by deriving an index theorem on the simple, but significant example of Heisenberg elliptic operators on a trivial foliation, which are in general not elliptic but hypoelliptic. The formula obtained is an extension of an index formula due to Fedosov.

The Splitting of the Domain of the Definition of the Elliptic Self-Adjoint Pseudodifferential Operator

In this work we researched domain splitting of self-adjoint elliptic pseudodifferential operator. In particular the Laplace -- Beltrami operator in the space of smooth differential $k$-forms defined on a smooth compact oriented Riemannian manifold without boundary be such operator. This result can be used in model with Sobolev type equations.

Local and nonlocal boundary conditions forμ-transmission and fractional elliptic pseudodifferential operators

A classical pseudodifferential operator $P$ on $R^n$ satisfies the $\mu$-transmission condition relative to a smooth open subset $\Omega $, when the symbol terms have a certain twisted parity on the normal to $\partial\Omega $. As shown recently by the author, the condition assures solvability of Dirichlet-type boundary problems for elliptic $P$ in full scales of Sobolev spaces with a singularity $d^{\mu -k}$, $d(x)=\operatorname{dist}(x,\partial\Omega)$. Examples include fractional Laplacians $(-\Delta)^a$ and complex powers of strongly elliptic PDE. We now introduce new boundary conditions, of Neumann type or more general nonlocal. It is also shown how problems with data on $R^n\setminus \Omega $ reduce to problems supported on $\bar\Omega$, and how the so-called "large" solutions arise. Moreover, the results are extended to general function spaces $F^s_{p,q}$ and $B^s_{p,q}$, including H\"older-Zygmund spaces $B^s_{\infty ,\infty}$. This leads to optimal H\"older estimates, e.g. for Dirichlet solutions of $(-\Delta)^au=f\in L_\infty (\Omega)$, $u\in d^aC^a(\bar\Omega)$ when $0<a<1$, $a\ne 1/2$ (in $d^aC^{a-\epsilon}(\bar\Omega)$ when $a=1/2$).

On the Index Formula for an Isometric Diffeomorphism

We give an elementary solution to the problem of the index of elliptic operators associated with shift operator along the trajectories of an isometric diffeomorphism of a smooth closed manifold. This solution is based on index-preserving reduction of the operator under consideration to some elliptic pseudo-differential operator on a higher-dimension manifold and on the application of the Atiyah–Singer formula. The final formula of the index is given in terms of the symbol of the operator on the original manifold.

$L^2$-index formula for proper cocompact group actions

We study index theory of G -invariant elliptic pseudo-differential operators acting on a complete Riemannian manifold, where a unimodular, locally compact group G acts properly, cocompactly and isometrically. An L^2 -index formula is obtained using the heat kernel method.

Betti numbers of random nodal sets of elliptic pseudo-differential operators

Given an elliptic self-adjoint pseudo-differential operator $P$ bounded from below, acting on the sections of a Riemannian line bundle over a smooth closed manifold $M$ equipped with some Lebesgue measure, we estimate from above, as $L$ grows to infinity, the Betti numbers of the vanishing locus of a random section taken in the direct sum of the eigenspaces of $P$ with eigenvalues below $L$. These upper estimates follow from some equidistribution of the critical points of the restriction of a fixed Morse function to this vanishing locus. We then consider the examples of the Laplace-Beltrami and the Dirichlet-to-Neumann operators associated to some Riemannian metric on $M$.

Spectral theory of a hybrid class of pseudo-differential operators

In this paper we introduce the hybrid class of the so-called -hypoelliptic symbols, and consider the corresponding pseudo-differential operators. With any -elliptic pseudo-differential operator with positive order, we associate the minimal and maximal operators on . Further on, we prove that the minimal and maximal operators are equal and we compute their domains in terms of a family of suitable Sobolev spaces. In the last section, we show that an -elliptic pseudo-differential operator is Fredholm. Moreover, we discuss the essential spectra of -elliptic pseudo-differential operators with suitable orders.

Heat kernel estimates for pseudodifferential operators, fractional Laplacians and Dirichlet-to-Neumann operators

The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(−t P) associated with a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The Poissonian bounds generalize those obtained for perturbations of fractional powers of the Laplacian. In the selfadjoint case, extensions to $${t \in{\mathbb C}_+}$$ are studied. In particular, our results apply to the Dirichlet-to-Neumann semigroup.

Reduction of the calculus of pseudodifferential operators on a noncompact manifold to the calculus on a compact manifold of doubled dimension

The paper is devoted to the exposition of results announced in [1]. We construct a reduction (following an idea of S. P. Novikov) of the calculus of pseudodifferential operators on Euclidean space ℝn to a similar calculus in the space of sections of a one-dimensional fiber bundle ξ on the 2n-dimensional torus \(\mathbb{T}^{2n} \). This reduction enables us to identify the Schwartz space on ℝn with the space of smooth sections Γ∞(T2n, ξ), compare the Sobolev norms on the corresponding spaces and pseudodifferential operators in them, and describe the class of elliptic operators that reduce to Fredholm operators in Sobolev norms. Thus, for a natural class of elliptic pseudodifferential operators on a noncompact manifold of ℝn, we construct an index formula in accordance with the classical Atya-Singer formula.

Pseudodifferential extension and Todd class

Let M be a closed manifold. Wodzicki shows that, in the stable range, the cyclic cohomology of the associative algebra of pseudodifferential symbols of order ≤0 is isomorphic to the homology of the cosphere bundle of M. In this article we develop a formalism which allows to calculate that, under this isomorphism, the Radul cocycle corresponds to the Poincaré dual of the Todd class. As an immediate corollary we obtain a purely algebraic proof of the Atiyah–Singer index theorem for elliptic pseudodifferential operators on closed manifolds.

Extended Sobolev Scale and Elliptic Operators

We obtain a constructive description of all Hilbert function spaces that are interpolation spaces with respect to a pair of Sobolev spaces $$ \left[ {{H^{{\left( {{s_0}} \right)}}}\left( {{{\mathbb{R}}^n}} \right),{H^{{\left( {{s_1}} \right)}}}\left( {{{\mathbb{R}}^n}} \right)} \right] $$ of some integer orders s 0 and s 1 and form an extended Sobolev scale. We propose equivalent definitions of these spaces with the use of uniformly elliptic pseudo-differential operators positive-definite in $$ {L_2}\left( {{{\mathbb{R}}^n}} \right) $$ . Possible applications of the introduced scale of spaces are indicated.

The spectral function of a first order elliptic system

We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of complex-valued half-densities over a connected compact manifold without boundary. The eigenvalues of the principal symbol are assumed to be simple but no assumptions are made on their sign, so the operator is not necessarily semi-bounded. We study the following objects: the propagator (time-dependent operator which solves the Cauchy problem for the dynamic equation), the spectral function (sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the manifold, with summation carried out over all eigenvalues between zero and a positive \lambda ), and the counting function (number of eigenvalues between zero and a positive \lambda ). We derive explicit two-term asymptotic formulae for all three. For the propagator "asymptotic'' is understood as asymptotic in terms of smoothness, whereas for the spectral and counting functions "asymptotic'' is understood as asymptotic with respect to \lambda\to+\infty .

Fredholmness property of $$M$$ -elliptic pseudo-differential operator under change variable in its symbol

This paper deals with minimal and maximal pseudo-differential operators. Moreover, it is given necessary and sufficient conditions for which these operators can be Fredholm. A description of their essential spectrum has been obtained for operators of non-zero order. Finally, it is shown that change variable preserve Fredholm property of pseudo-differential operators.

An Index Formula for Perturbed Dirac Operators on Lie Manifolds

We prove an index formula for a class of Dirac operators coupled with unbounded potentials, also called “Callias-type operators”. More precisely, we study operators of the form $P := \hspace* {.5mm} / \hspace* {-2.3mm}D+ V$ , where $\hspace* {.5mm} / \hspace* {-2.3mm}D$ is a Dirac operator and V is an unbounded potential at infinity on a non-compact manifold M 0. We assume that M 0 is a Lie manifold with compactification denoted by M. Examples of Lie manifolds are provided by asymptotically Euclidean or asymptotically hyperbolic spaces and many others. The potential V is required to be such that V is invertible outside a compact set K and V −1 extends to a smooth vector bundle endomorphism over M∖K that vanishes on all faces of M in a controlled way. Using tools from analysis on non-compact Riemannian manifolds, we show that the computation of the index of P reduces to the computation of the index of an elliptic pseudodifferential operator of order zero on M 0 that is a multiplication operator at infinity. The index formula for P can then be obtained from the results of Carvalho (in K-theory 36(1–2):1–31, 2005). As a first step in the proof, we obtain a similar index formula for general pseudodifferential operators coupled with bounded potentials that are invertible at infinity on a restricted class of Lie manifolds, so-called asymptotically commutative, which includes, for instance, the scattering and double-edge calculi. Our results extend many earlier, particular results on Callias-type operators.

Global existence and blow-up of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations

This study deals with the analysis of the Cauchy problem of a general class of nonlocal nonlinear equations modeling the bi-directional propagation of dispersive waves in various contexts. The nonlocal nature of the problem is reflected by two different elliptic pseudodifferential operators acting on linear and nonlinear functions of the dependent variable, respectively. The well-known doubly dispersive nonlinear wave equation that incorporates two types of dispersive effects originated from two different dispersion operators falls into the category studied here. The class of nonlocal nonlinear wave equations also covers a variety of well-known wave equations such as various forms of the Boussinesq equation. Local existence of solutions of the Cauchy problem with initial data in suitable Sobolev spaces is proven and the conditions for global existence and finite-time blow-up of solutions are established.

Classical and quantum ergodicity on orbifolds

We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdi\`ere and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quantum ergodicity for the operator P. We also prove ergodicity of the geodesic flow on a compact Riemannian orbifold of negative sectional curvature.