Vector-valued multiparameter singular integrals and pseudodifferential operators

We consider multiparameter singular integrals and pseudodifferential operators acting on mixed-norm Bochner spaces Lp₁,…,pN(Rn₁×?×RnN;X) where X is a UMD Banach space satisfying Pisier's property (α). These geometric conditions are shown to be necessary. We obtain a vector-valued version of a result by R. Fefferman and Stein, also providing a new, inductive proof of the original scalar-valued theorem. Then we extend a result of Bourgain on singular integrals in UMD spaces with an unconditional basis to a multiparameter situation. Finally we carry over a result of Yamazaki on pseudodifferential operators to the Bochner space setting, improving the known vector-valued results even in the one-parameter case.     

Complex powers of hypoelliptic pseudodifferential operators

Complex powers of a class of hypoelliptic pseudodifferential operators in Rn, as well as their heat kernels are studied. An application to the Schatten-von Neumann property of pseudodifferential operators is given.     

Projective pseudodifferential analysis and harmonic analysis

We consider pseudodifferential operators on functions on Rn+1 which commute with the Euler operator, and can thus be restricted to spaces of functions homogeneous of some given degree. The symbols of such restrictions can be regarded as functions on a reduced phase space, isomorphic to the homogeneous space Gn/Hn=SL(n+1,R)/GL(n,R), and the resulting calculus is a pseudodifferential analysis of operators acting on spaces of appropriate sections of line bundles over the projective space Pn(R): these spaces are the representation spaces of the maximal degenerate series (πiλ,ε) of Gn. This new approach to the quantization of Gn/Hn, already considered by other authors, has several advantages: as an example, it makes it possible to give a very explicit version of the continuous part from the decomposition of L²(Gn/Hn) under the quasiregular action of Gn. We also consider interesting special symbols, which arise from the consideration of the resolvents of certain infinitesimal operators of the representation πiλ,ε.     

Fundamental solutions of homogeneous elliptic differential operators

We compute fundamental solutions of homogeneous elliptic differential operators, with constant coefficients, on Rn by mean of analytic continuation of distributions. The result obtained is valid in any dimension, for any degree and can be extended to pseudodifferential operators of the same type.     

Spectral theory of some degenerate elliptic operators with local singularities

This paper is based on our previous results (Haroske and Skrzypczak (2008) [23], Haroske and Skrzypczak (in press) [25]) on compact embeddings of Muckenhoupt weighted function spaces of Besov and Triebel-Lizorkin type with example weights of polynomial growth near infinity and near some local singularity. Our approach also extends (Haroske and Triebel (1994) [21]) in various ways. We obtain eigenvalue estimates of degenerate pseudodifferential operators of type b₂○p(x,D)○b₁ where bi∈Lri(Rn,wi), wi∈A_∞, i=1,2, and , ?>0. Finally we deal with the ‘negative spectrum’ of some operator Hγ=A−γV for γ→∞, where the potential V may have singularities (in terms of Muckenhoupt weights), and A is a positive elliptic pseudodifferential operator of order ?>0, self-adjoint in L₂(Rn). This part essentially relies on the Birman-Schwinger principle. We conclude this paper with a number of examples, also comparing our results with preceding ones.     

The metaplectic representation,Weyl operators and spectral theory

Let X be the Grassmannian of Lagrangian subspaces of R²ⁿ and π: Θ → X the bundle of negative half-forms. We construct a canonical imbedding S(Rⁿ)_even → C^∞(Θ) which intertwines the metaplectic representation of M_p(n) on S(Rⁿ) with the induced representation of M_p(n) on C^∞(Θ). This imbedding converts the algebra of Weyl operators into an algebra of pseudodifferential operators and enables us to prove theorems about the spectral properties of Weyl operators by reducing them to standard facts about pseudodifferential operators. For instance we are able to prove a Weyl theorem on the asymptotic growth of eigenvalues with an “optimal” error estimate for such operators and an analogue of the Helton clustering theorem and the Chazarain-Duistermaat-Guillemin trace formula.     

Characterization of pseudodifferential operators in Hölder-Zygmund spaces

We consider pseudodifferential operators with symbols of the Hörmander class S _{1, δ} ^m , 0 ≤ δ < 1, in Hölder-Zygmund spaces on ? ⁿ and obtain a Beals-type characterization of such operators. By way of application, we show that the inverse of a pseudodifferential operator invertible in a Hölder-Zygmund space is itself a pseudodifferential operator, and hence, the spectra of a pseudodifferential operator in the space L ₂ and in the Hölder-Zygmund spaces coincide as sets.     

Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II. The compact group case

Let G⊂O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A₀ in L²(Rn) that commutes with the regular representation of G, and assume that it is elliptic on X. We show that the spectrum of the Friedrichs extension A of the operator is discrete, and using the method of the stationary phase, we derive asymptotics for the number Nχ(λ) of eigenvalues of A equal or less than λ and with eigenfunctions in the χ-isotypic component of L²(X) as λ→∞, giving also an estimate for the remainder term for singular group actions. Since the considered critical set is a singular variety, we recur to partial desingularization in order to apply the stationary phase theorem.     

Calcul fonctionnel a plusieurs variables pour des operateurs pseudodifferentiels dansR n

We study aC ^∞ functional calculus with several variables forv pseudodifferential operatorsP ₁, …,P _v inR ⁿ . Whenf is a function belonging to the classS _1.0 ^r (R ^v ) of Hörmander, we prove that, under some conditions,f(P ₁,…,P _v) is a pseudodifferential operator, and we give an asymptotic formula for its symbol.     

Reduced Weyl asymptotics for pseudodifferential operators on bounded domains I. The finite group case

Let G⊂O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A₀ in L²(Rn) with G-invariant Weyl symbol, and assume that it is semi-bounded from below. We show that the spectrum of the Friedrichs extension A of the operator is discrete, and derive asymptotics for the number Nχ(λ) of eigenvalues of A less or equal λ and with eigenfunctions in the χ-isotypic component of L²(X) as λ→∞, giving also an estimate for the remainder term in case that G is a finite group. In particular, we show that the multiplicity of each unitary irreducible representation in L²(X) is asymptotically proportional to its dimension.     

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 ? ⁿ 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 ? ⁿ with the space of smooth sections Γ^∞(T ²ⁿ , 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 ? ⁿ , we construct an index formula in accordance with the classical Atya-Singer formula.     

Products of longitudinal pseudodifferential operators on flag varieties

Associated to each set S of simple roots of SL(n,C) is an equivariant fibration X→XS of the complete flag variety X of Cn. To each such fibration we associate an algebra JS of operators on L²(X), or more generally on L²-sections of vector bundles over X. This ideal contains, in particular, the longitudinal pseudodifferential operators of negative order tangent to the fibres. Together, they form a lattice of operator ideals whose common intersection is the compact operators. Thus, for instance, the product of negative order pseudodifferential operators along the fibres of two such fibrations, X→XS and X→XT, is a compact operator if S∪T is the full set of simple roots. The construction of the ideals uses noncommutative harmonic analysis, and hinges upon a representation theoretic property of subgroups of SU(n), which may be described as ‘essential orthogonality of subrepresentations’.     

Small conjugacy classes in the automorphism groups of relatively free groups

Let F be an infinitely generated free group and let R be a fully invariant subgroup of F such that (a) R is contained in the commutator subgroup F^′ of F and (b) the quotient group F/R is residually torsion-free nilpotent. Then the automorphism group of the group F/R^′ is complete. In particular, the automorphism group of any infinitely generated free solvable group of derived length at least two is complete.This extends a result by Dyer and Formanek (1977) [7] on finitely generated groups F_n/R^′ where F_n is a free group of finite rank n at least two and R a characteristic subgroup of F_n.     

Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces

In this paper we study continuity and invertibility of pseudodifferential operators with non-regular Banach space valued symbols. The corresponding pseudodifferential operators generate analytic semigroups on the Sobolev spaces W _p ^k (? ⁿ , E) with k ∈ ?₀, 1 ≤ p ≤ ∞. Here E is an arbitrary Banach space. We also apply the theory to solve non-autonomous parabolic pseudodifferential equations in Sobolev spaces.     

Orthogonal resolutions of lines in AG(n,q)

A resolution of the lines of AG(n,q) is a partition of the lines classes (called resolution classes) such that every point of the geometry is on exactly one line of each resolution class. Two resolutions R,R' of AG(n,q) are orthogonal if any resolution class from R has at most one line in common with any class from R'. In this paper, we construct orthogonal resolutions on AG(n,q) for all n=2ⁱ⁺¹, i=1,2,…, and all q>2 a prime power. The method involves constructing AG(n,q) from a finite projective plane of order q^n-1 and using the structure of the plane to display the orthogonal resolutions.     

Sharp estimates for a class of hyperbolic pseudo-differential equations

In this paper we consider the Cauchy problem for a class of hyperbolic pseudodifferential operators. The considered class contains constant coefficient differential equations, also allowing the coefficients to depend on time. We establish sharp L ^p ? L^p, Lipschitz, and other estimates for their solutions. In particular, the ellipticity condition for the roots of the principal symbol is eliminated for certain dimensions. We discuss the situation with no loss of smoothness for solutions. In the space R¹⁺ⁿ with n ≤ 4 (total dimension ≤ 5), we give a complete list of L ^p ? L^p properties. In particular, this contains the very important case R¹⁺³.     

Mixed exterior Laplace's problem

In [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator. An approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1997) 55-81], authors study Dirichlet and Neumann problems for the Laplace operator in exterior domains of Rn. This paper extends this study to the resolution of a mixed exterior Laplace's problem. Here, we give existence, unicity and regularity results in Lp's theory with 1<p<∞, in weighted Sobolev spaces.     

Coercive singular perturbations II: Reduction and convergence

An algebra of pseudodifferential singular perturbations is introduced. It provides a constructive machinery in order to reduce an elliptic singularly perturbed operator (in $\text{R}$ⁿ or on a smooth manifold without border) to a regular perturbation. The technique developed is applied to some singularly perturbed boundary value problems as well. Special attention is given to a singular perturbation appearing in the linear theory of thin elastic plates. A Wiener-Hopf-type operator containing the small parameter reduces this singular perturbation to a regular one. It also gives rise to a natural recurrence process for the construction of high-order asymptotic formulae for the solution of the perturbed problem. The method presented can be extended to the general coercive singular perturbations.     

Inequalities for integral means over symmetric sets

We prove that the integral of n functions over a symmetric set L in Rn, with additional properties, increases when the functions are replaced by their symmetric decreasing rearrangements. The result is known when L is a centrally symmetric convex set, and our result extends it to nonconvex sets. We deduce as consequences, inequalities for the average of a function whose level sets are of the same type as L, over measurable sets in Rn. The average of such a function on E is maximized by the average over the symmetric set E^*.     

On the maximal L p -regularity of parabolic mixed-order systems

We study maximal L _p -regularity for a class of pseudodifferential mixed-order systems on a space–time cylinder \mathbbRⁿ ×\mathbbR{\mathbb{R}^n \times \mathbb{R}} or X ×\mathbbR{X \times \mathbb{R}} , where X is a closed smooth manifold. To this end, we construct a calculus of Volterra pseudodifferential operators and characterize the parabolicity of a system by the invertibility of certain associated symbols. A parabolic system is shown to induce isomorphisms between suitable L _p -Sobolev spaces of Bessel potential or Besov type. If the cross section of the space–time cylinder is compact, the inverse of a parabolic system belongs to the calculus again. As applications, we discuss time-dependent Douglis–Nirenberg systems and a linear system arising in the study of the Stefan problem with Gibbs–Thomson correction.