Elliptic Pseudodifferential Operators Research Articles

Estimates for Sums of Eigenfunctions of Elliptic Pseudo-differential Operators on Compact Lie Groups

We extend the estimates proved by Donnelly and Fefferman, and by Lebeau, Robbiano and Zuazua, for sums of eigenfunctions of the Laplacian (on a compact manifold) to estimates for sums of eigenfunctions of any positive and elliptic pseudo-differential operator of positive order on a compact Lie group. Our criteria are imposed in terms of the positivity of the corresponding matrix-valued symbol of the operator. As an application of these inequalities in control theory, we obtain the null-controllability for diffusion models for elliptic pseudo-differential operators on compact Lie groups.

The logarithmic Dirichlet Laplacian on Ahlfors regular spaces

We introduce the logarithmic analogue of the Laplace-Beltrami operator on Ahlfors regular metric-measure spaces. This operator is intrinsically defined with spectral properties analogous to those of elliptic pseudo-differential operators on Riemannian manifolds. Specifically, its heat semigroup consists of compact operators which are trace-class after some critical point in time. Moreover, its domain is a Banach module over the Dini continuous functions and every Hölder continuous function is a smooth vector. Finally, the operator is compatible, in the sense of noncommutative geometry, with the action of a large class of non-isometric homeomorphisms.

Unconditional convergence of eigenfunction expansions for abstract and elliptic operators

We study the most general class of eigenfunction expansions for abstract normal operators with pure point spectrum in a complex Hilbert space. We find sufficient conditions for such expansions to be unconditionally convergent in spaces with two norms and also estimate the degree of this convergence. Our result essentially generalizes and complements the known theorems of Krein and of Krasnosel'skiĭ and Pustyl'nik. We apply it to normal elliptic pseudodifferential operators on compact boundaryless $C^{\infty }$ -manifolds. We find generic conditions for eigenfunction expansions induced by such operators to converge unconditionally in the Sobolev spaces $W^{\ell }_{p}$ with $p>2$ or in the spaces $C^{\ell }$ (specifically, for the $p$ -th mean or uniform convergence on the manifold). These conditions are sufficient and necessary for the indicated convergence on Sobolev or Hörmander function classes and are given in terms of parameters characterizing these classes. We also find estimates for the degree of the convergence on such function classes. These results are new even for differential operators on the circle and for multiple Fourier series.

Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases

Let $P$ be a symmetric $2a$-order classical strongly elliptic pseudodifferential operator with \emph{even} symbol $p(x,\xi)$ on $\mathbb{R}^n $ ($0<a<1$), for example a perturbation of $(-\Delta )^a$. Let $\Omega \subset \mathbb{R}^n$ be bounded, and let $P_D$ be the Dirichlet realization in $L_2(\Omega)$ defined under the exterior condition $u=0$ in $\mathbb{R}^n\setminus\Omega$. When $p(x,\xi)$ and $\Omega$ are $C^\infty $, it is known that the eigenvalues $\lambda_j$ (ordered in a nondecreasing sequence for $j\to \infty$) satisfy a Weyl asymptotic formula \begin{equation*} \lambda _j(P_{D})=C(P,\Omega )j^{2a/n}+o(j^{2a/n}) \text {for $j\to \infty $}, \end{equation*} with $C(P,\Omega)$ determined from the principal symbol of $P$. We now show that this result is valid for more general operators with a possibly nonsmooth $x$-dependence, over Lipschitz domains, and that it extends to $\tilde P=P+P'+P”$, where $P'$ is an operator of order $<\min\{2a, a+\frac 12\}$ with certain mapping properties, and $P”$ is bounded in $L_2(\Omega )$ (e.g. $P”=V(x)\in L_\infty(\Omega)$). Also the regularity of eigenfunctions of $P_D$ is discussed.

Weighted local Weyl laws for elliptic operators

Let $A$ be an elliptic pseudo-differential operator of order $m$ on a closed manifold $\man$ of dimension $n>0$, formally positive self-ajdoint with respect to some positive smooth density $d\mu_\man$. Then, the spectrum of $A$ is made up of a sequence of eigenvalues $(\lambda_k)_{k\geq 1}$ whose corresponding eigenfunctions $(e_k)_{k\geq 1}$ are $C^\infty$ smooth. Fix $s\in\R$ and define \ \[ K_L^s(x,y)=\sum_{0<\lambda_k\leq L}\lambda_k^{-s} e_k(x)\overline{e_k(y)}\, . \] We derive asymptotic formulae near the diagonal for the kernels $K_L^s(x,y)$ when $L\rightarrow +\infty$ with fixed $s$. For $s=0$, $K^0_L$ is the kernel of the spectral projector studied by H\ormander in [Hor68]. In the present work we build on H\ormander's result to study the kernels $K^s_L$. If $s<\frac{n}{m}$, $K_L^s$ is of order $L^{-s+n/m}$ and near the diagonal, the rescaled leading term behaves like the Fourier transform of an explicit function of the symbol of $A$. If $s=\frac{n}{m}$, under some explicit generic condition on the principal symbol of $A$, which holds if $A$ is a differential operator, the kernel has order $\ln(L)$ and the leading term has a logarithmic divergence smoothed at scale $L^{-1/m}$. Our results also hold for elliptic differential Dirichlet eigenvalue problems.

On Certain Operator Families

In this paper, we propose an abstract scheme for the study of special operators and apply this scheme to examining elliptic pseudo-differential operators and related boundary-value problems on manifolds with nonsmooth boundaries. In particular, we consider cases where boundaries may contain conical points, edges of various dimensions, and even peak points. Using the constructions proposed, we present well-posed formulations of boundary-value problems for elliptic pseudo-differential equations on manifolds discussed in Sobolev–Slobodecky spaces.

Ellipticity and Fredholmness of pseudo-differential operators on ℓ²(ℤⁿ)

The minimal operator and the maximal operator of an elliptic pseudo-differential operator with symbols on Z n × T n \mathbb {Z}^n\times \mathbb {T}^n are proved to coincide and the domain is given in terms of a Sobolev space. Ellipticity and Fredholmness are proved to be equivalent for pseudo-differential operators on Z n \mathbb {Z}^n . The index of an elliptic pseudo-differential operator on Z n \mathbb {Z}^n is also computed.

Invariant subspaces of elliptic systems II: Spectral theory

Consider an elliptic self-adjoint pseudodifferential operator A acting on m -columns of half-densities on a closed manifold M M, whose principal symbol is assumed to have simple eigenvalues.We show that the spectrum of A decomposes, up to an error with superpolynomial decay, into m distinct series, each associated with one of the eigenvalues of the principal symbol of A . These spectral results are then applied to the study of propagation of singularities in hyperbolic systems. The key technical ingredient is the use of the carefully devised pseudodifferential projections introduced in the first part of this work, which decompose L^2 (M) into almost-orthogonal almost-invariant subspaces under the action of both A and the hyperbolic evolution.

Invariant subspaces of elliptic systems I: Pseudodifferential projections

Consider an elliptic self-adjoint pseudodifferential operator A acting on m-columns of half-densities on a closed manifold M, whose principal symbol is assumed to have simple eigenvalues. We show existence and uniqueness of m orthonormal pseudodifferential projections commuting with the operator A and provide an algorithm for the computation of their full symbols, as well as explicit closed formulae for their subprincipal symbols. Pseudodifferential projections yield a decomposition of L2(M) into invariant subspaces under the action of A modulo C∞(M). Furthermore, they allow us to decompose A into m distinct sign definite pseudodifferential operators. Finally, we represent the modulus and the Heaviside function of the operator A in terms of pseudodifferential projections and discuss physically meaningful examples.

Parametric pseudodifferential operators with point-singularity in the covariable

Starting out from a new description of a class of parameter-dependent pseudodifferential operators with finite regularity number due to G. Grubb, we introduce a calculus of parameter-dependent, poly-homogeneous symbols whose homogeneous components have a particular type of point-singularity in the covariable-parameter space. Such symbols admit intrinsically a second kind of expansion which is closely related to the expansion in the Grubb–Seeley calculus and permits to recover the resolvent-trace expansion for elliptic pseudodifferential operators originally proved by Grubb–Seeley. Another application is the invertibility of parameter-dependent operators of Toeplitz type, i.e., operators acting in subspaces determined by zero-order pseudodifferential idempotents.

Diagonalization of elliptic systems via pseudodifferential projections

Consider an elliptic self-adjoint pseudodifferential operator A acting on m-columns of half-densities on a closed manifold M, whose principal symbol is assumed to have simple eigenvalues. Relying on a basis of pseudodifferential projections commuting with A, we construct an almost-unitary pseudodifferential operator that diagonalizes A modulo an infinitely smoothing operator. We provide an invariant algorithm for the computation of its full symbol, as well as an explicit closed formula for its subprincipal symbol. Finally, we give a quantitative description of the relation between the spectrum of A and the spectrum of its approximate diagonalization, and discuss the implications at the level of spectral asymptotics.

Addendum to “The equivariant spectral function of an invariant elliptic operator. Lp-bounds, caustics, and concentration of eigenfunctions” [J. Math. Pures Appl. 120 (2018) 33–90

Let M be a compact boundaryless Riemannian manifold, carrying an effective and isometric action of a torus T, and P0 an invariant elliptic classical pseudodifferential operator on M. In this note, we strengthen the asymptotics for the equivariant (or reduced) spectral function of P0 derived in [6], which are already sharp in the eigenvalue aspect, to become almost sharp in the isotypic aspect. In particular, this leads to hybrid equivariant Lp-bounds for eigenfunctions that are almost sharp in the eigenvalue and isotypic aspect.

Integration by parts for nonsymmetric fractional-order operators on a halfspace

For a strongly elliptic pseudodifferential operator L of order 2a (0<a<1) with real kernel, we show an integration-by-parts formula for solutions of the homogeneous Dirichlet problem, in the model case where the operator is x-independent with homogeneous symbol, considered on the halfspace R+n. The new aspect compared to (−Δ)a is that L is nonsymmetric, having both an even and an odd part. Hence it satisfies a μ-transmission condition where generally μ≠a.We present a complex method, relying on a factorization in factors holomorphic in ξn in the lower or upper complex halfplane, using order-reducing operators combined with a decomposition principle originating from Wiener and Hopf. This is in contrast to a real, computational method presented very recently by Dipierro, Ros-Oton, Serra and Valdinoci. Our method allows μ in a larger range than they consider.Another new contribution is the (model) study of “large” solutions of nonhomogeneous Dirichlet problems when μ>0. Here we deduce a “halfways Green's formula” for L:∫R+nLuv¯dx−∫R+nuL⁎v‾dx=c∫Rn−1γ0(u/xnμ−1)γ0(v¯/xnμ⁎)dx′, when u solves a nonhomogeneous Dirichlet problem for L, and v solves a homogeneous Dirichlet problem for L⁎; μ⁎=2a−μ. Finally, we show a full Green's formula, when both u and v solve nonhomogeneous Dirichlet problems; here both Dirichlet and Neumann traces of u and v enter, as well as a first-order pseudodifferential operator over the boundary.

Calderón problem for Yang–Mills connections

We consider the problem of identifying a unitary Yang–Mills connection \nabla on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian \nabla^*\nabla over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the case of trivial line bundles in the smooth category and for the higher rank case in the analytic category, by using geometric analysis methods. Moreover, by using a Runge-type approximation argument along curves to recover holonomy, we are able to uniquely determine both the bundle structure and the connection. Also, we prove that the DNmap is an elliptic pseudodifferential operator of order one on the restriction of the vector bundle to the boundary, whose full symbol determines the complete Taylor series of an arbitrary connection, metric and an associated potential at the boundary.

Functional calculus for elliptic operators on noncommutative tori, I

In this paper, we introduce a parametric pseudodifferential calculus on noncommutative n-tori which is a natural nest for resolvents of elliptic pseudodifferential operators. Unlike in some previous approaches to parametric pseudodifferential calculi, our parametric pseudodifferential calculus contains resolvents of elliptic pseudodifferential operators that need not be differential operators. As an application we show that complex powers of positive elliptic pseudodifferential operators on noncommutative n-tori are pseudodifferential operators. This confirms a claim of Fathi-Ghorbanpour-Khalkhali.

The Inverse Problem of Magneto-Electroencephalography is Well-Posed: it has a Unique Solution that is Stable with Respect to Perturbations

Contrary to the opinion that has prevailed for the last several decades about the incorrectness of the inverse–MEEG problems (see, for example, the paper of D. Sheltraw and E. Coutsias in Journal of Applied Physics, 94, No. 8, 5307–5315 (2003)), in this note it is shown that this problem is absolutely well posed: it has a unique solution, but in a special class of functions (different from those considered by biophysicists). The solution has the form q = q0 + p0δ|∂Y, where q0 is an ordinary function defined in the domain of the region Y occupied by the brain, and p0δ|∂Y is a δ-function on the boundary of the domain Y with a certain density p0. Moreover, the operator of this problem realizes an isomorphism of the corresponding function spaces. This result was obtained due to the fact that: (1) Maxwell’s equations are taken as a basis; (2) a transition was made to the equations for the potentials of the magnetic and electric fields; (3) the theory of boundary value problems for elliptic pseudodifferential operators with an entire index of factorization is used. This allowed us to find the correct functional class of solutions of the corresponding integral equation of the first kind. Namely: the solution has a singular boundary layer in the form of a delta function (with some density) at the boundary of the domain. From the point of view of the MEEG problem, this means that the sought-for current dipoles are also concentrated in the cerebral cortex.

Well-posedness of the EPDiff equation with a pseudo-differential inertia operator

In this article we study the class of right-invariant, fractional order Sobolev-type metrics on groups of diffeomorphisms of a compact manifold M. Our main result concerns well-posedness properties for the corresponding Euler-Arnold equations, also called the EPDiff equations, which are of importance in mathematical physics and in the field of shape analysis and template registration. Depending on the order of the metric, we will prove both local and global well-posedness results for these equations. As a result of our analysis we will also obtain new commutator estimates for elliptic pseudo-differential operators.

Operator Symbols and Operator Indices

We suggest a certain variant of symbolic calculus for special classes of linear bounded operators acting in Banach spaces. According to the calculus we formulate an index theorem and give applications to elliptic pseudo-differential operators on smooth manifolds with non-smooth boundaries.

Projective elliptic genera and elliptic pseudodifferential genera

In this paper, we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle. Such projective elliptic genera are rational q-series that have topological definition and also have analytic interpretation via the fractional index theorem in [25] without requiring spin condition. We prove the modularity properties of these projective elliptic genera. As an application, we construct elliptic pseudodifferential genera for any elliptic pseudodifferential operator. This suggests the existence of putative S1-equivariant elliptic pseudodifferential operators on loop space whose equivariant indices are elliptic pseudodifferential genera.

A Nash–Moser approach for the Euler–Arnold equations

We study the local well-posedness in the smooth category for a class of Euler equations. A Nash–Moser approach is used to extend, for the case of an invertible elliptic pseudo-differential operator, some results obtained by Escher and Kolev, with the help of some geometric arguments.