Class Of Pseudodifferential Operators Research Articles

Geometry of pseudodifferential algebra bundles and Fourier integral operators

We study the geometry and topology of (filtered) algebra bundles ΨZ over a smooth manifold X with typical fiber ΨZ(Z;V), the algebra of classical pseudodifferential operators acting on smooth sections of a vector bundle V over the compact manifold Z and of integral order. First, a theorem of Duistermaat and Singer is generalized to the assertion that the group of projective invertible Fourier integral operators PG(FC(Z;V)) is precisely the automorphism group of the filtered algebra of pseudodifferential operators. We replace some of the arguments in their work by microlocal ones, thereby removing the topological assumption. We define a natural class of connections and B-fields on the principal bundle to which ΨZ is associated and obtain a de Rham representative of the Dixmier–Douady class in terms of the outer derivation on the Lie algebra and the residue trace of Guillemin and Wodzicki. The resulting formula only depends on the formal symbol algebra ΨZ/Ψ−∞. Examples of pseudodifferential algebra bundles are given that are not associated to a finite-dimensional fiber bundle over X.

Elliptic operators on refined Sobolev scales on vector bundles

Abstract We introduce a refined Sobolev scale on a vector bundle over a closed infinitely smooth manifold. This scale consists of inner product Hörmander spaces parametrized with a real number and a function varying slowly at infinity in the sense of Karamata. We prove that these spaces are obtained by the interpolation with a function parameter between inner product Sobolev spaces. An arbitrary classical elliptic pseudodifferential operator acting between vector bundles of the same rank is investigated on this scale. We prove that this operator is bounded and Fredholm on pairs of appropriate Hörmander spaces. We also prove that the solutions to the corresponding elliptic equation satisfy a certain a priori estimate on these spaces. The local regularity of these solutions is investigated on the refined Sobolev scale. We find new sufficient conditions for the solutions to have continuous derivatives of a given order.

Spectral functions of subordinate Brownian motion on closed manifolds:

For a class of subordinators, we investigate the spectrum of the infinitesimal generator of subordinate Brownian motion on a closed manifold. We consider three spectral functions of the generator: the zeta function, the heat trace and the spectral action. Each spectral function explicitly yields both probabilistic and geometric information, the latter through the classical heat invariants. All constructions are done with classical pseudodifferential operators and are fully analytically tractable.

Integration by parts and Pohozaev identities for space-dependent fractional-order operators

Consider a classical elliptic pseudodifferential operator P on Rn of order 2a(0<a<1) with even symbol. For example, P=A(x,D)a where A(x,D) is a second-order strongly elliptic differential operator; the fractional Laplacian (−Δ)a is a particular case. For solutions u of the Dirichlet problem on a bounded smooth subset Ω⊂Rn, we show an integration-by-parts formula with a boundary integral involving (d−au)|∂Ω, where d(x)=dist(x,∂Ω). This extends recent results of Ros-Oton, Serra and Valdinoci, to operators that are x-dependent, nonsymmetric, and have lower-order parts. We also generalize their formula of Pohozaev-type, that can be used to prove unique continuation properties, and nonexistence of nontrivial solutions of semilinear problems. An illustration is given with P=(−Δ+m2)a. The basic step in our analysis is a factorization of P, P∼P−P+, where we set up a calculus for the generalized pseudodifferential operators P± that come out of the construction.

Lp bounds in S(m, g)-calculus

Given a self-adjoint second-order differential operator with positive characteristic. In this paper, we establish estimates for a class of pseudo-differential operators associated to in the setting of -calculus without subellipticity assumptions on the operator .

Heat Trace Asymptotics of Subordinate Brownian Motion in Euclidean Space

For a class of Laplace exponents we derive the heat trace asymptotics of the generator of the corresponding subordinate Brownian motion on Euclidean space. The terms in the asymptotic expansion are found to depend both on the geometry of Euclidean space and probabilistic properties of the subordinator. The key assumption is the existence of a suitable density for the Levy measure of the subordinator. An intermediate step is the computation of the zeta function of the generator. We employ methods from the theory of classical pseudodifferential operators on Euclidean space. The analysis is highly explicit and fully analytically tractable.

On bounded pseudodifferential operators in Wiener spaces

We aim at extending the definition of the Weyl calculus to an infinite dimensional setting, by replacing the phase space R2n by B2, where (i,H,B) is an abstract Wiener space. A first approach is to generalize the integral definition using the Wigner function. The symbol is then a function defined on B2 and belonging to a L1 space for a Gaussian measure, the Weyl operator is defined as a quadratic form on a dense subspace of L2(B). For example, the symbol can be the stochastic extension on B2, in the sense of L. Gross, of a function F which is continuous and bounded on H2. In the second approach, this function F defined on H2 satisfies differentiability conditions analogous to the finite dimensional ones. One needs to introduce hybrid operators acting as Weyl operators on the variables of finite dimensional subset of H and as anti-Wick operators on the rest of the variables. The final Weyl operator is then defined as a limit and it is continuous on a L2 space. Under rather weak conditions, it is an extension of the operator defined by the first approach. We give examples of monomial symbols linking this construction to the classical pseudodifferential operators theory and other examples related to other fields or previous works on this subject.

On the Hörmander–Mihlin theorem for mixed-norm Lebesgue spaces

For some questions in the theory of partial differential equations, it is useful to consider the mixed-norm Lebesgue spaces introduced by Benedek and Panzone (1962). We show that Hörmander's results (1960) obtained on translation invariant operators on Lebesgue spaces extend to the mixed-norm case. We also revisit the Hörmander–Mihlin multiplier theorem in this setting, which was first obtained by Lizorkin (1970), where we particularly consider the form of the constants used in the estimates. This improvement allows us to prove the boundedness of classical pseudodifferential operators on mixed-norm Lebesgue spaces, as well as generalising the H-distributions, a recently introduced (2011, by Mitrović and the first author) extension of H-measures, to these spaces.

The canonical trace and the noncommutative residue on the noncommutative torus

Using a global symbol calculus for pseudodifferential operators on tori, we build a canonical trace on classical pseudodifferential operators on noncommutative tori in terms of a canonical discrete sum on the underlying toroidal symbols. We characterise the canonical trace on operators on the noncommutative torus as well as its underlying canonical discrete sum on symbols of fixed (resp. any) noninteger order. On the grounds of this uniqueness result, we prove that in the commutative setup, this canonical trace on the noncommutative torus reduces to Kontsevich and Vishik’s canonical trace which is thereby identified with a discrete sum. A similar characterisation for the noncommutative residue on noncommutative tori as the unique trace which vanishes on trace–class operators generalises Fathizadeh and Wong’s characterisation in so far as it includes the case of operators of fixed integer order. By means of the canonical trace, we derive defect formulae for regularized traces. The conformal invariance of the ζ \zeta –function at zero of the Laplacian on the noncommutative torus is then a straightforward consequence.

Reduction of nonlocal pseudodifferential operators on a noncompact manifold to classical pseudodifferential operators on a double-dimensional compact manifold

In the present paper, we obtain some Fredholmness criteria and a formula for the index of nonlocal pseudodifferential operators generated by shifts and multiplications by periodic functions and acting in the Schwartz space S(ℝn). These results significantly differ from the results known to the author in this field of study, namely, the Fredholmness criteria and the formula for the index of nonlocal pseudodifferential operators acting over the noncompact manifold ℝn are obtained here for the first time.

Hadamard States for the Linearized Yang–Mills Equation on Curved Spacetime

We construct Hadamard states for the Yang-Mills equation linearized around a smooth, space-compact background solution. We assume the spacetime is globally hyperbolic and its Cauchy surface is compact or equal $\rr^d$. We first consider the case when the spacetime is ultra-static, but the background solution depends on time. By methods of pseudodifferential calculus we construct a parametrix for the associated vectorial Klein-Gordon equation. We then obtain Hadamard two-point functions in the gauge theory, acting on Cauchy data. A key role is played by classes of pseudodifferential operators that contain microlocal or spectral type low-energy cutoffs. The general problem is reduced to the ultra-static spacetime case using an extension of the deformation argument of Fulling, Narcowich and Wald. As an aside, we derive a correspondence between Hadamard states and parametrices for the Cauchy problem in ordinary quantum field theory.

On some properties of a class of degenerate pseudodifferential operators

A new class of pseudodifferential operators with degeneration is considered. The operators are constructed using a special integral transform mapping a weighted differentiation operator to a multiplication operator. The composition and boundedness properties of such operators in special weighted spaces are examined. Theorems on commutation of such operators with differentiation operators are obtained. The behavior of these operators as t → 0and t → +∞ is investigated. The properties of adjoint operators are studied, and an analogue of Garding’s inequality is proved.

The broken ray transform in $n$ dimensions with flat reflecting boundary

We study the broken ray transform on $n$-dimensional Euclidean domains where the reflecting parts of the boundary are flat and establish injectivity and stability under certain conditions. Given a subset $E$ of the boundary $\partial \Omega$ such that $\partial \Omega \setminus E$ is itself flat (contained in a union of hyperplanes), we measure the attenuation of all broken rays starting and ending at $E$ with the standard optical reflection rule applied to $\partial \Omega \setminus E$. By localizing the measurement operator around broken rays which reflect off a fixed sequence of flat hyperplanes, we can apply the analytic microlocal approach of Frigyik, Stefanov, and Uhlmann for the ordinary ray transform by means of a local path unfolding. This generalizes the author's previous result for the square, although we can no longer treat reflections from corner points. Similar to the result for the two dimensional square, we show that the normal operator is a classical pseudo differential operator of order -1 plus a smoothing term with $C_{0}^{\infty}$ Schwartz kernel.

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 $a=1/2$).

Fractional Laplacians on domains, a development of Hörmander's theory of μ-transmission pseudodifferential operators

Let P be a classical pseudodifferential operator of order m∈C on an n-dimensional C∞ manifold Ω1. For the truncation PΩ to a smooth subset Ω there is a well-known theory of boundary value problems when PΩ has the transmission property (preserves C∞(Ω¯)) and is of integer order; the calculus of Boutet de Monvel. Many interesting operators, such as for example complex powers of the Laplacian (−Δ)μ with μ∉Z, are not covered. They have instead the μ-transmission property defined in Hörmander's books, mapping xnμC∞(Ω¯) into C∞(Ω¯). In an unpublished lecture note from 1965, Hörmander described an L2-solvability theory for μ-transmission operators, departing from Vishik and Eskin's results. We here develop the theory in Lp Sobolev spaces (1<p<∞) in a modern setting. It leads to not only Fredholm solvability statements but also regularity results in full scales of Sobolev spaces (s→∞). The solution spaces have a singularity at the boundary that we describe in detail. We moreover obtain results in Hölder spaces, which radically improve recent regularity results for fractional Laplacians.

The Broken Ray Transform on the Square

We study a particular broken ray transform on the Euclidean unit square and establish injectivity and stability for \(C_{0}^{2}\) perturbations of a vanishing absorption parameter \(\sigma \equiv 0\). Given an open subset \(E\) of the boundary, we measure the attenuation of all broken rays starting and ending at \(E\) with the standard optical reflection rule applied to \(\partial \Omega {\setminus } E\). Using the analytic microlocal approach of Frigyik et al. for the X-ray transform on generic families of curves, we show injectivity via a path unfolding argument under suitable conditions on the available broken rays. Then we show that with a suitable decomposition of the measurement operator via smooth cutoff functions, the associated normal operator is a classical pseudo differential operator of order \(-1\), which leads to the desired result.

Global functional calculus for operators on compact Lie groups

In this paper we develop the functional calculus for elliptic operators on compact Lie groups without the assumption that the operator is a classical pseudo-differential operator. Consequently, we provide a symbolic descriptions of complex powers of such operators. As an application, we give a constructive symbolic proof of the Gårding inequality for operators in (ρ,δ)-classes in the setting of compact Lie groups.

Hörmander Class of Pseudo-Differential Operators on Compact Lie Groups and Global Hypoellipticity

Abstract In this paper we give several global characterisations of the Hörmander class $$\Psi ^m(G)$$ Ψ m ( G ) of pseudo-differential operators on compact Lie groups in terms of the representation theory of the group. The result is applied to give criteria for the ellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of the first and second order globally hypoelliptic differential operators are given, in particular of operators that are locally not invertible nor hypoelliptic but globally are. Where the global hypoelliptiticy fails, one can construct explicit examples based on the analysis of the global symbols.

Singular equivariant asymptotics and Weyl’s law. On the distribution of eigenvalues of an invariant elliptic operator

Abstract We study the spectrum of an invariant, elliptic, classical pseudodifferential operator on a closed Riemannian manifold M carrying an effective and isometric action of a compact, connected Lie group G. Using resolution of singularities, we determine the asymptotic distribution of eigenvalues along the isotypic components, and relate it to the reduction of the corresponding Hamiltonian flow, proving that the reduced spectral counting function satisfies Weyl’s law, together with an estimate for the remainder.

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.