pinecone/core-2020-05-10-deduplication huggingface dataset:
On the Meromorphic Extension of the Spherical Functions on Noncompactly Causal Symmetric Spaces

AbstractWe determine integral formulas for the meromorphic extension in the λ-parameter of the spherical functions ϕλ on a noncompactly causal symmetric space. The main tool is Bernstein's theorem on the meromorphic extension of complex powers of polynomials. The regularity properties of ϕλ are deduced. In particular, the possible λ-poles of ϕλ are located among the translates of the zeros of the Bernstein polynomial. The translation parameter depends only on the structure of the symmetric space. The expression of the Bernstein polynomial is conjectured. The relation between the Bernstein polynomial and the product formula of the cΩ-function is analyzed. The conjecture is verified in the rank-one case. The explicit formulas obtained in this case yield a detailed description of singularities of ϕλ. In the general higher rank case, the integral formulas are applied to find asymptotic estimates for the spherical functions. In the Appendix, the spherical functions on noncompactly causal symmetric spaces are regarded as a special instance of Harish-Chandra-type expansions associated with roots systems with arbitrary multiplicities. We study expansions obtained by taking averages over arbitrary parabolic subgroups of the Weyl group of the root system. The possible λ-singularities are located in this general context
Generalized Ornstein–Uhlenbeck Semigroups: Littlewood–Paley–Stein Inequalities and the P. A. Meyer Equivalence of Norms

AbstractLet μ be a nondegenerate Gaussian measure on a Hilbert space H. For an arbitrary selfadjoint nonnegative operator A we consider the semigroup etL=Γ(e−tA) on Lp(μ), where Γ stands for the second quantization operator. We provide an explicit characterization of the domains of (I−L)m/2 in Lp(μ) in terms of Gaussian Sobolev spaces thus extending the P. A. Meyer result on equivalence of norms. The main tools are the Littlewood–Paley–Stein inequalities which are proved under minimal assumptions by a purely analytic method following E. Stein (1970, “Topics in Harmonic Analysis,” Princeton Univ. Press, Princeton, NJ)
Local Contractions of Banach Spaces and Spectral Gap Conditions

AbstractWe study the linearization problem for a Ck, 1, k⩾1, contraction of a Banach space E near a fixed point which satisfies a spectral gap condition and a narrow band condition both of order k. We also assume that the part of the spectrum in each band satisfies a finite non-resonant condition of order k relative to itself together with the part that lies in the larger bands. We show that there is a Ck, β linearization for sufficiently small β>0. We give a precise estimate on β in terms of the gap and band conditions
Random Holonomy for Hopf Fibrations

AbstractIn this paper we study the heat flow on sections of the complex and quaternion line bundles associated to the Hopf fibration over complex and quaternionic projective space. These bundles are equipped with natural metrics and connections. We represent solutions to the heat-equation by the expectation of an appropriate stochastic functional on path space. This uses the stochastic parallel transport. We obtain the first eigenvalue of the horizontal Laplacian by calculating conditional expectations of the stochastic parallel transport. We use a stochastic argument similar to a skew-product decomposition to reduce the problem to an ordinary differential equation of Sturm–Liouville type. We find all eigenvalues and eigenfunctions of this equation
Matricial Nehari Problems, J-Inner Matrix Functions and the Muckenhoupt Condition

AbstractThe classes of regular and strongly regular γ-generating matrices and J-inner matrix valued functions arose in the investigation of the matricial Nehari problem, bitangential interpolation problems, and inverse problems for canonical systems as well as the theory of characteristic functions of operators and operator nodes. In this paper, new characterizations of these classes are developed. In particular, the property of strong regularity is characterized in terms of a matricial Muckenhoupt (A2) condition in the Treil–Volberg form. These results are based on parametrizations that are intimately connected with Darlington representations of matrix valued functions in the Schur and Carathéodory classes. As a byproduct of this analysis, examples of strongly regular γ-generating matrices and entire J-inner matrix valued functions that are unbounded on the circle and the real line, respectively, are presented
Optimal Existence Theorems for Nonhomogeneous Differential Inclusions

AbstractIn this paper we address the question of solvability of the differential inclusions (1.1). Our approach to these problems is based on the idea of constructing a sequence of approximate solutions which converges strongly and makes use of Gromov's idea (following earlier work of Nash and Kuiper) to control convergence of the gradients by appropriate selection of the elements of the sequence. In this paper we identify an optimal setting of this method. In particular we show that the existence result holds for general upper semicontinuous functions H without extra requirements like quasiconvexity of H with respect to Du, which was assumed in previous works, where the idea to apply the Baire category lemma to the sets of approximate solutions was developed. We also apply our result to identify the minimal sets, where the function H should vanish to guarantee solvability of the inclusions
Integral Representations and Liouville Theorems for Solutions of Periodic Elliptic Equations

AbstractThe paper contains integral representations for certain classes of exponentially growing solutions of second order periodic elliptic equations. These representations are the analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation. When one restricts the class of solutions further, requiring their growth to be polynomial, one arrives to Liouville type theorems, which describe the structure and dimension of the spaces of such solutions. The Liouville type theorems previously proved by M. Avellaneda and F.-H. Lin and J. Moser and M. Struwe for periodic second order elliptic equations in divergence form are significantly extended. Relations of these theorems with the analytic structure of the Fermi and Bloch surfaces are explained
A Strong Maximum Principle for the Laplace Equation with Mixed Boundary Condition

AbstractIn this work we present a comparison result for two solutions of the Laplace equation in a smooth bounded domain, satisfying the same mixed boundary condition (zero Dirichlet data on part of the boundary and zero Neumann data on the rest). The result is in some sense a generalization of the Hopf lemma to the case of mixed boundary conditions, where the barrier function is not given explicitly, but as the solution of the Laplace equation with a constant right hand side and mixed boundary conditio
A Krein-like Formula for Singular Perturbations of Self-Adjoint Operators and Applications

AbstractGiven a self-adjoint operator A:D(A)⊆H→H and a continuous linear operator τ:D(A)→X with Range τ′∩H′={0}, X a Banach space, we explicitly construct a family AτΘ of self-adjoint operators such that any AτΘ coincides with the original A on the kernel of τ. Such a family is obtained by giving a Kreın-like formula where the role of the deficiency spaces is played by the dual pair (X, X′); the parameter Θ belongs to the space of symmetric operators from X′ to X. When X=C one recovers the “H−2-construction” of Kiselev and Simon and so, to some extent, our results can be regarded as an extension of it to the infinite rank case. Considering the situation in which H=L2(Rn) and τ is the trace (restriction) operator along some null subset, we give various applications to singular perturbations of non necessarily elliptic pseudo-differential operators, thus unifying and extending previously known results
The Shilov Boundary of an Operator Space and the Characterization Theorems

AbstractWe study operator spaces, operator algebras, and operator modules from the point of view of the noncommutative Shilov boundary. In this attempt to utilize some noncommutative Choquet theory, we find that Hilbert C*-modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We introduce certain multiplier operator algebras and C*-algebras of an operator space, which generalize the algebras of adjointable operators on a C*-module and the imprimitivity C*-algebra. It also generalizes a classical Banach space notion. This multiplier algebra plays a key role here. As applications of this perspective, we unify and strengthen several theorems characterizing operator algebras and modules. We also include some general notes on the commutative case of some of the topics we discuss, coming in part from joint work with Christian Le Merdy, about function modules
Global Bounds of Schrödinger Heat Kernels with Negative Potentials

AbstractWe obtain global in time bounds for the heat kernel G of the Schrödinger operator L=−Δ+V. The potential V satisfies V(x)∼−C/d(x)b near infinity with b∈(0, ∞). The result can be described as follows. Suppose L is positive and b=2. Then G=G(x, t; y, 0) has a global upper bound which is a standard Gaussian times a polynomial function of x, y, t. However when L is not nonnegative and negative eigenvalues exist G=G(x, t; 0, 0) is bounded below by a standard Gaussian times ectwithc>0. In other words, the growth rate of G(x, t; 0, 0) is comparable with the heat kernel of −Δ−c, regardless how fast V decays near infinity
Analyticity and Discrete Maximal Regularity on Lp-Spaces

AbstractIn the first part of this paper, we give the following interpolation result on the analyticity (i.e. the property ‖(T−I)Tn‖⩽C/n for all n∈N) of an operator T on Lp: If T is powerbounded on Lp and Lq as well as analytic on Lp, then T is powerbounded and analytic on Lr for all r strictly between p and q. This is a discrete analogue of the well-known corresponding result for analytic semigroups (etA). As recently shown by the author, the analyticity of T is a necessary condition for the maximal regularity of the discrete time evolution equation un+1−Tun=fn for all n∈Z+, u0=0. In the second part of this paper we establish the following two sufficient conditions for its maximal regularity: T is a subpositive analytic contraction, or T is an integral operator satisfying certain Poisson bounds. These results are discrete analogues of the corresponding results for the maximal regularity of the evolution equation u′(t)−Au(t)=f(t) for all t∈R+, u(0)=0, due to Lamberton, Weis, Coulhon and Duong and Hieber and Prüss. For the Poisson bound result of Coulhon and Duong and Hieber and Prüss we give a slight improvement and a short proof
Scattering Theory for the Dirac Operator with a Long-Range Electromagnetic Potential

AbstractWe consider the Dirac operator with a long-range potential V(x). Scalar, pseudo-scalar and vector components of V(x) may have arbitrary power-like decay at infinity. We introduce wave operators with time-independent modifiers. These modifiers are pseudo-differential operators whose symbols are, roughly speaking, constructed in terms of approximate eigenfunctions of the stationary problem. We derive and solve eikonal and transport equations for the corresponding phase and amplitude functions. From an analytical point of view, our proof of the existence and completeness of the wave operators relies on the limiting absorption principle and radiation estimates established in the paper. This allows us to fit the long-range scattering theory for the Dirac operator into the framework of smooth perturbations. Finally, we find the asymptotics for large times t of solutions u(x, t) of the time-dependent Dirac equation
Asgeirsson's Mean Value Theorem and Related Identities

AbstractAsgeirsson's mean value theorem states that if u satisfies the ultrahyperbolic differential equation (Δx−Δy)u=0 in a neighborhood of the convex compact set KR={(x, y); x, y∈Rν, |x|+|y|⩽R}, then 〈u, f〉=0 where f is the difference between the surface measures on the spheres {(x, 0); x∈Rν, |x|=R} and {(0, y); y∈Rν, |y|=R}. We extend this to solutions of the inhomogeneous equation by proving that f=(Δx−Δy)μR where μR=12χ(1−ν)/2+(πAR/4R2), with AR(x, y)=(|x|2−|y|2)2−2R2 (|x|2+|y|2)+R4. The distributions χa+ on R are defined by χa+(t)=ta+/Γ(a+1) when Rea>−1 and then continued analytically to all a∈C. This formula is closely related to the fundamental solution of the wave equation in Rν+1. Similar identities are given for arbitrary indefinite nonsingular real quadratic forms in R2ν. This work was originally motivated by the progressive solutions of the wave equation given by G. Friedlander and M. Riesz
The Structure of the Resolvent of Elliptic Pseudodifferential Operators

AbstractWe show that the resolvent kernel of an elliptic b-pseudodifferential operator on a compact manifold with corners (of arbitrary codimension) is a polyhomogeneous, or classical, function on a certain manifold with corners. The singularities of the resolvent kernel are shown to localize near the diagonal as the resolvent parameter goes to infinity. Explicit descriptions of the expansions, including logarithmic terms, are given. In particular, the asymptotics of the resolvent restricted to the diagonal follows as a corollary. Applications to the asymptotic behavior of the heat kernel and to the analysis of the poles of complex powers are also given
Heat Equation Derivative Formulas for Vector Bundles

AbstractWe use martingale methods to give Bismut type derivative formulas for differentials and co-differentials of heat semigroups on forms, and more generally for sections of vector bundles. The formulas are mainly in terms of Weitzenböck curvature terms; in most cases derivatives of the curvature are not involved. In particular, our results improve B. K. Driver's formula in (1997, J. Math. Pures Appl. (9)76, 703–737) for logarithmic derivatives of the heat kernel measure on a Riemannian manifold. Our formulas also include the formulas of K. D. Elworthy and X.-M. Li (1998, C. R. Acad. Sci. Paris Sér. I Math.327, 87–92)
Fourier Analysis on Starlike Lipschitz Surfaces

AbstractA theory of a class of singular integrals on starlike Lipschitz surfaces in Rn is established. The class of singular integrals forms an operator algebra identical to the class of bounded holomorphic Fourier multipliers, as well as to the Cauchy–Dunford bounded holomorphic functional calculus of the spherical Dirac operator. The study proposes a new method inducing Clifford holomorphic functions from holomorphic functions of one complex variable, by means of which problems on the sphere are reduced to those on the unit circle
On Boundary Value Problems for Dirac Type Operators I. Regularity and Self-Adjointness

AbstractIn a series of papers, we will develop systematically the basic spectral theory of (self-adjoint) boundary value problems for operators of Dirac type. We begin in this paper with the characterization of (self-adjoint) boundary conditions with optimal regularity, for which we will derive the heat asymptotics and index theorems in subsequent publications. Along with a number of new results, we extend and simplify the proofs of many known theorems. Our point of departure is the simple structure which is displayed by Dirac type operators near the boundary. Thus our proofs are given in an abstract functional analytic setting, generalizing considerably the framework of compact manifolds with boundary. The results of this paper have been announced previously by the authors (J. Brüning and M. Lesch, in “Geometric Aspects of Partial Differential Equations (B. Booss- Bavnbek and K. P. Wojciechowski, Eds.), Contemporary Mathematics, Vol. 242, pp. 203–205, Amer. Math. Soc., Providence, RI, 1999)
Functional Calculus for the Ornstein–Uhlenbeck Operator

AbstractLet γ be the Gauss measure on Rd and L the Ornstein–Uhlenbeck operator, which is self adjoint in L2(γ). For every p in (1, ∞), p≠2, set φ*p=arcsin|2/p−1| and consider the sector Sφ*p={z∈C:|argz|<φ*p}. The main result of this paper is that if M is a bounded holomorphic function on Sφ*p whose boundary values on ∂Sφ*p satisfy suitable Hörmander type conditions, then the spectral operator M(L) extends to a bounded operator on Lp(γ) and hence on Lq(γ) for all q such that |1/q−1/2|⩽|1/p−1/2|. The result is sharp, in the sense that L does not admit a bounded holomorphic functional calculus in a sector smaller than Sφ*p
Extensions of Operator Valued Positive Definite Functions and Commutant Lifting on Ordered Groups

AbstractLet Ω be a locally compact abelian ordered group. We say that Ω has the extension property if every operator valued continuous positive definite function on an interval of Ω has a positive definite extension to the whole group and we say that Ω has the commutant lifting property if a natural extension of the commutant lifting theorem holds on Ω. We give a characterization of the groups having the extension property in terms of unitary extensions of a particular class of multiplicative family of partial isometries. It is proved that if a group has the extension property and satisfies an archimedean condition then it has the commutant lifting property. It is also proved that if the ordered group Γ has the extension property and satisfies an archimedean condition then Ω=Γ×Z with the lexicographic order has the extension property. As an application we obtain that the groups Zn and R×Zn with the lexicographic order have the extension property and the commutant lifting property
Maximal, Minimal, and Primary Invariant Subspaces

AbstractLet X be a complex infinite dimensional Banach space. An operator L on X is called of subcritical class, if ∑∞n=1n−3/2log+‖Ln‖<∞. Assume that T is an operator on X whose iterates have norms of polynomial growth. We prove that if T has a range of finite codimension and a left inverse of subcritical class, then every maximal invariant subspace of T has codimension one, and if T has a finite dimensional kernel and a right inverse of subcritical class, then every minimal invariant subspace of T is one dimensional. Using these results we obtain new information on the invariant subspace lattices of shifts and backward shifts on a wide class of Banach spaces of analytic functions on the unit disc. We also introduce the notion of primary invariant subspaces, and determine their structure for a large class of shifts
Propagation of Singularities in Many-Body Scattering in the Presence of Bound States

AbstractIn this paper we describe the propagation of singularities of tempered distributional solutions u∈S′ of (H−λ)u=0, where H is a many-body Hamiltonian H=Δ+V, Δ⩾0, V=∑aVa, and λ is not a threshold of H, under the assumption that the inter-particle (e.g., two-body) interactions Va are real-valued polyhomogeneous symbols of order −1 (e.g., Coulomb-type with the singularity at the origin removed). Here the term “singularity” refers to a microlocal description of the lack of decay at infinity. Thus, we prove that the set of singularities of u is a union of maximally extended broken bicharacteristics of H. These are curves in the characteristic variety of H−λ, which can be quite complicated due to the existence of bound states. We use this result to describe the wave front relation of the S-matrices. We also analyze Lagrangian properties of this relation, which shows that the relation is not “too large” in terms of its dimension
An Infinite-Dimensional Analogue of the Lebesgue Measure and Distinguished Properties of the Gamma Process

AbstractWe define a one-parameter family Lθ of sigma-finite (finite on compact sets) measures in the space of distributions. These measures are equivalent to the laws of the classical gamma processes and invariant under an infinite-dimensional abelian group of certain positive multiplicators. This family of measures was first discovered by Gelfand–Graev–Vershik in the context of the representation theory of current groups; here we describe it in direct terms using some remarkable properties of the gamma processes. We show that the class of multiplicative measures coincides with the class of zero-stable measures which is introduced in the paper. We give also a new construction of the canonical representation of the current group SL(2, R)X
Connections and Curvature in the Riemannian Geometry of Configuration Spaces

AbstractTorsion free connections and a notion of curvature are introduced on the infinite dimensional nonlinear configuration space Γ of a Riemannian manifold M under a Poisson measure. This allows us to state identities of Weitzenböck type and energy identities for anticipating stochastic integral operators. The one-dimensional Poisson case itself gives rise to a non-trivial geometry, a de Rham–Hodge–Kodaira operator, and a notion of Ricci tensor under the Poisson measure. The methods used in this paper have been thus far applied to d-dimensional Brownian path groups and rely on the introduction of a particular tangent bundle and associated damped gradient
Scattering Theory for Conformally Compact Metrics with Variable Curvature at Infinity

AbstractWe develop the scattering theory of a general conformally compact metric by treating the Laplacian as a degenerate elliptic operator (with non-constant indicial roots) on a compact manifold with boundary. Variability of the roots implies that the resolvent admits only a partial meromorphic continuation, and the bulk of the paper is devoted to studying the structure of the resolvent, Poisson, and scattering kernels for frequencies outside the region of meromorphy. For low frequencies the scattering matrix is shown to be a pseudodifferential operator with frequency dependent domain. In particular, generalized eigenfunctions exhibit L2 decay in directions where the asymptotic curvature is sufficiently negative. We explicitly construct the resolvent kernel for generic frequency in this part of the continuous spectrum
Magnetic Bottles in Connection with Superconductivity

AbstractMotivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, a lot of papers devoted to the analysis in a semi-classical regime of the lowest eigenvalue of the Schrödinger operator with magnetic field have appeared recently. Here we mention the works by Bernoff–Sternberg, Lu–Pan and Del Pino–Felmer–Sternberg. This partially recovers questions analyzed in a different context by the authors around the question of the so called magnetic bottles. Our aim is to analyze the former results, to treat them in a more systematic way and to improve them by giving sharper estimates of the remainder. In particular, we improve significantly the lower bounds and as a by-product we solve a conjecture proposed by Bernoff–Sternberg concerning the localization of the ground state inside the boundary in the case with constant magnetic fields