Periodic Pseudo-differential Operators Research Articles

Determinants and Plemelj-Smithies formulas

We establish Plemelj-Smithies formulas for determinants in different algebras of operators. In particular we define a Poincaré type determinant for operators on the torus {{mathbb T}^n} and deduce formulas for determinants of periodic pseudo-differential operators in terms of the symbol. On the other hand, by applying a recently introduced notion of invariant operators relative to fixed decompositions of Hilbert spaces we also obtain formulae for determinants with respect to the trace class. The analysis makes use of the corresponding notion of full matrix-symbol. We also derive explicit formulas for determinants associated to elliptic operators on compact manifolds, compact Lie groups, and on homogeneous vector bundles over compact homogeneous manifolds.

Two-dimensional Dirac operators with singular interactions supported on closed curves

We study the two-dimensional Dirac operator with a class of interface conditions along a smooth closed curve, which model the so-called electrostatic and Lorentz scalar interactions of constant strengths, and we provide a rigorous description of their self-adjoint realizations and their qualitative spectral properties. We are able to cover in a uniform way all so-called critical combinations of coupling constants, for which there is a loss of regularity in the operator domain. For the case of a non-zero mass term, this results in an additional point in the essential spectrum, which reflects the creation of an infinite number of eigenvalues in the central gap, and the position of this point can be made arbitrary by a suitable choice of the parameters. The analysis is based on a combination of the extension theory of symmetric operators with a detailed study of boundary integral operators viewed as periodic pseudodifferential operators.

$$L^p$$-Boundedness and $$L^p$$-Nuclearity of Multilinear Pseudo-differential Operators on $${\mathbb {Z}}^n$$ and the Torus $${\mathbb {T}}^n$$

In this article, we begin a systematic study of the boundedness and the nuclearity properties of multilinear periodic pseudo-differential operators and multilinear discrete pseudo-differential operators on $L^p$-spaces. First, we prove analogues of known multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Grafakos, Tomita, Torres, Kenig, Stein, Fujita, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. Later, we investigate the $s$-nuclearity, $0<s \leq 1,$ of periodic and discrete pseudo-differential operators. To accomplish this, we classify those $s$-nuclear multilinear integral operators on arbitrary Lebesgue spaces defined on $\sigma$-finite measures spaces. We also study similar properties for periodic Fourier integral operators. Finally, we present some applications of our study to deduce the periodic Kato-Ponce inequality and to examine the $s$-nuclearity of multilinear Bessel potentials as well as the $s$-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.

On the boundedness of periodic pseudo-differential operators

In this paper we investigate the \(L^p\)-boundedness of certain classes of periodic pseudo-differential operators. The operators considered arise from the study of symbols on \({\mathbb {T}}^n\times {\mathbb {Z}}^n\) with limited regularity.

Hölder–Besov boundedness for periodic pseudo-differential operators

In this work we give Holder–Besov estimates for periodic Fourier multipliers. We present a class of bounded pseudo-differential operators on periodic Besov spaces with symbols of limited regularity.

Weak type (1,1) bounds for a class of periodic pseudo-differential operators

In this work we establish the weak (1,1) continuity for pseudo-differential operators with symbols in toroidal $$(1,\delta )$$ classes.

Representations of almost periodic pseudodifferential operators and applications in spectral theory

The paper concerns algebras of almost periodic pseudodifferential operators on \({\mathbb{R}^d}\) with symbols in Hormander classes. We study three representations of such algebras, one of which was introduced by Coburn, Moyer and Singer and the other two inspired by results in probability theory by Gladyshev. Two of the representations are shown to be unitarily equivalent for nonpositive order. We apply the results to spectral theory for almost periodic pseudodifferential operators acting on L2 and on the Besicovitch Hilbert space of almost periodic functions.

Almost periodic pseudodifferential operators and Gevrey classes

We study almost periodic pseudodifferential operators acting on almost periodic functions \({G_{\rm ap}^s(\mathbb {R}^d)}\) of Gevrey regularity index s ≥ 1. We prove that almost periodic operators with symbols of Hörmander type \({S_{\rho,\delta}^m}\) satisfying an s-Gevrey condition are continuous on \({G_{\rm ap}^s(\mathbb {R}^d)}\) provided 0 < ρ ≤ 1, δ = 0 and s ρ ≥ 1. A calculus is developed for symbols and operators using a notion of regularizing operator adapted to almost periodic Gevrey functions and its duality. We apply the results to show a regularity result in this context for a class of hypoelliptic operators.

On the Toroidal Quantization of Periodic Pseudo-Differential Operators

On the torus, pseudo-differential operators can be presented globally by Fourier series, without local coordinate charts. Periodization of partial differential equations leads to investigating the correspondence between toroidal and Euclidean symbols of operators. Fourier series operators—the analogues of Fourier integral operators—are introduced, and formulae for their compositions with pseudo-differential operators are presented. The L 2 boundedness of pseudo-differential and Fourier series operators follows from certain conditions on phases and amplitudes.

A Lidskii type formula for Dixmier traces

We present a new formula to compute Dixmier traces τ ω ( T ) of pseudodifferential operators (respectively, almost periodic pseudodifferential operators) of order − n on n-dimensional compact Riemannian manifolds (respectively, R n ). Under a natural condition on the operator T, we show that τ ω ( T ) = ω - lim t → ∞ 1 log ( 1 + t ) ∫ λ ∉ 1 t G λ d μ T ( λ ) , where G is any bounded neighborhood of 0 ∈ C and μ T is the Brown spectral measure of T. If T is measurable, then the ω-limit may be replaced with the true (ordinary) limit. Our approach works equally well in both type I and II settings. To cite this article: N.A. Azamov, F.A. Sukochev, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Commutator Characterization of Periodic Pseudodifferential Operators

We show in a novel way that periodic pseudodifferential operators are pseudodiffer-ential operators in Hörmander’s definition. In our approach, commutators a la Beals, Dunau, Coifman and Meyer on \mathbb R^n and on closed manifolds are involved.

An Integral Operator Representation of Classical Periodic Pseudodifferential Operators

In this note we prove that every classical 1-periodic pseudodifferential operator A of order \alpha \in \mathbb R \\mathbb N_0 can be represented in the form (Au)(t) = \int^1_0 [\kappa_{\alpha}^+(t - s)a_+(t,s) + \kappa_{\alpha}^– (t-s)a\_(t,s) + a(t,s)]u(s)ds where \alpha_± and a are C^{\infty} -smooth 1-periodic functions and \kappa_{\alpha}^± are 1-periodic functions or distributions with Fourier coefficients \kappa_{\alpha}^+(n) = |n|^{\alpha} and \kappa_{\alpha}^–(n) = |n|^{\alpha} sign (n) (0 \neq n \in \mathbb Z) with respect to the trigonometric orthonormal basis \{e^{in2xt}\}_{n \in \mathbb Z} of L^2 (0,1) . Some explicit formulae for \kappa_{\alpha}^± are given. The case of operators of order \alpha \in \mathbb N_0 is discussed, too.

On Symbol Analysis of Periodic Pseudo differential Operators

This paper deals with periodic pseudodifferential operators on the unit circle \mathbb T^1 = \mathbb R^1/ \mathbb Z^1 . The main results are asymptotic expansions for adjoints and products of periodic pseudodifferential operators.

Local and Global Descriptions of Periodic Pseudodifferential Operators

AbstractPeriodic pseudodifferential operators can be defined either globally, via FOURIER series, or else locally, via partitions of unity and FOURIER integrals. Here, it is proved that the two definitions are equivalent.

C*-algebras of almost periodic pseudo-differential operators

Our basic goal is to develop an index theory for almost periodic pseudo-differential operators on R ~. The prototype of this theory is [5] which has direct application to the almost periodic Toeplitz operators. Here, we s tudy index theory for a C*-algebra of operators on R ~ which contains most almost periodic pseudo-differential operators such as those arising in the s tudy of elliptic boundary value problems for constant coefficient elliptic operators on a half space with almost periodic boundary conditions. Our program is as follows: We begin with a discussion of a C*-algebra with symbol which contains all of the classical pseudo-differential operators on R ~. Precisely, if A is a bounded operator on L2(R ~) and 2 ER ~, let e~(A) denote the conjugate of A with the function e *~'x acting as a multiplier denoted e~. We first s tudy the C*-algebra of those A for which the function 2~+e~(A) has a strongly continuous extension to the radial compactification of R ~. The restriction of this function to the complement of R ~ then gives the usual (principal) symbol a(A) when A is a pseudo-differential operator of order zero (of a suitable type). We characterize the Fourier multipliers in this algebra and the image of the symbol map. We give sufficient conditions for the usual construction of a pseudo-differential operator as well as one of Friedrichs' constructions to give an element of this algebra. In particular, the lat ter gives a positive linear right inverse for the symbol m a p a t least when the symbol is sufficiently smooth. I n fact, we show in w 3 tha t the Friedrichs map is a right inverse to the symbol map in the almost periodic case. We expect this to be true in the general case also.