pinecone/core-2020-05-10-deduplication huggingface dataset:
On Functions of Finite Baire Index

AbstractIt is proved that every function of finite Baire index on a separable metric spaceKis aD-function, i.e., a difference of bounded semi-continuous functions onK. In fact it is a strongD-function, meaning it can be approximated arbitrarily closely inD-norm, by simpleD-functions. It is shown that if thenth derived set ofKis non-empty for all finiten, there existD-functions onKwhich are not strongD-functions. Further structural results for the classes of finite index functions and strongD-functions are also given
McKennon-Type Spaces for the Hankel Transformation

AbstractForμ>−\frac12;, the Zemanian space Hμof Hankel-transformable functions is expressed as the projective limit of a chain {Spμ}p∈Nof reflexive Banach spaces where the Hankel transformation Sμis an isometric isomorphism. Moreover, Hμis dense in Spμ(p∈N) and Spμis dense in Sqμ(p, q, ∈N, p⩾q). The images of these results in the dual spaces then follow. A fundamental rôle in the construction is played by the space S0μ, consisting of all thoseφ=φ(t) (t∈I=]0, ∞[) such that bothφand Sμφlie in the spaceLμof functions Lebesgue integrable onIwith respect to the weighttμ+1/2. We show that, under Hankel convolution, S0μis a Banach algebra and a dense ideal ofLμ, namely that formed by the functions inLμwhich are Hankel–Stieltjes transforms of Radon measuresγonIwith the property that ∫∞0tμ+1/2d|γ|(t)<∞. New results on the inversion of the Hankel transformation and the positivity of certain Hankel transforms are also established
Spherical Functions on Ol'shanskii Spaces

AbstractOl'shanskii spaces for semisimple groups are special cases of ordered symmetric spaces. The theory of spherical functions on semisimple Ol'shanskii spaces is extended to general Ol'shanskii spaces. The detailed structure theory for Lie algebras with invariant cones is used to generalize geometric results for Ol'shanskii spaces. Spherical functions are then defined via a set of integral equations using a Volterra algebra that consists ofG-invariant kernels satisfying a causality condition. It is shown that the resulting integral spherical formulas suffice to build a spherical Laplace transform and an inversion formula for this transform
Extensions and Contractions of the Lie Algebra ofq-Pseudodifferential Symbols on the Circle

AbstractWe construct cocycles on the Lie algebra of pseudo- andq-pseudodifferential symbols of one variable and on their close relatives: the sine-algebra and the Poisson algebra on two-torus. A “quantum” Godbillon–Vey cocycle on (pseudo)-differential operators appears in this construction as a natural generalization of the Gelfand–Fuchs 3-cocycle on periodic vector fields. A nontrivial embedding of the Virasoro algebra into (a completion of)q-pseudodifferential symbols is proposed. We describeq-analogs of the KP and KdV-hierarchies admitting an infinite number of conserved charges as well asq-deformed Gelfand–Dickey structures
The Dimension of the Brownian Frontier Is Greater Than 1

AbstractConsider a planar Brownian motion run for finite time. Thefrontieror “outer boundary” of the path is the boundary of the unbounded component of the complement. K. Burdzy (Prob. Theory Related Field83(1989), 135–205) showed that the frontier has infinite length. We improve this by showing that the Hausdorff dimension of the frontier is strictly greater than 1. (It has been conjectured that the Brownian frontier has dimension 4/3, but this is still open.) The proof uses Jones's Traveling Salesman Theorem and a self-similar tiling of the plane by fractal tiles known as Gosper Islands
The Inverse Segal–Bargmann Transform for Compact Lie Groups

AbstractIn this paper I give a new inversion formula (Theorem 1) for the generalized Segal–Bargmann transform introduced in B. C. Hall,J. Funct. Anal.122(1994), 103–151. The inversion formula may be viewed as a formula for theinverseheat operator for a compact Lie group. I then use this formula to give a new direct proof of the unitary of theK-invariant form of the Segal–Bargmann transform (Theorem 2). The proof of the inversion formula relies on an identity (Theorem 5) which relates the Laplacian for a compact Lie groupKto the Laplacian for the non-compact dual symmetric space KC/K
Traces of Sobolev Functions on Fractal Type Sets and Characterization of Extension Domains

AbstractWe describe traces of Sobolev functionsu∈W1, p(Rn), 1<p⩽∞, on certain subsets of Rnin terms of Sobolev spaces on metric spaces [7]. Our results apply to smooth submanifolds, fractal subsets, as well as to open subsets of Rn. In particular if 0⊂Rnis a John domain, then we characterize thoseW1, p(Ω) functions which can be extended toW1, p(Rn). IfΩis uniform, then this result implies Jones' extension theorem [14]. In the case of traces on fractal subsets our results are related to those of Jonsson and Wallin [16]
Restrictions of Unitary Representations to Lattices and AssociatedC*-Algebras

AbstractLetGbe a connected simple Lie group with finite centre, and letΓbe a lattice inG. Letπbe an irreducible unitary representation ofG, and letπ∣1 be the restriction ofπtoΓ. A result by Cowling and Steger states thatπ∣1 remains irreducible ifπis not a discrete series representation and thatπ∣1 is determined byπ. Our first result shows that even the weak equivalence class ofπ∣1 is determined byπwhenπis a complementary series representation. LetC*π(Γ) denote theC*-algebra generated by allπ(γ) forγinΓ. We show thatC*π(Γ) has a unique maximal two-sided ideal and a unique normalized trace
Asymptotics of Matrix Coefficients and Closures of Fourier–Stieltjes Algebras

AbstractThe vanishing theorems of R. Howe and C. C. Moore are interpreted in a topological way. This yields a generalization of their results to connected locally compact totally minimal groups. The first section gives the complete classification of these groups and shows a lot of examples. It is proven that, among the connected groups, the totally minimal groups are exactly those groupsGwhose Fourier–Stieltjes algebras are uniformly dense in the C*-algebra of all weakly almost periodic functions WAP(G). This solves a problem which was intensely discussed by C. Chou. Furthermore it is shown that WAP(G) is generated as a C*-algebra by the continuous functions onG/Nwhich vanish at infinity, whereNranges over all closed normal subgroups
The Construction of Filtrations on Abstract Wiener Space

AbstractThe abstract Wiener space, unlike the classical Wiener space, does not possess a natural notion of time. The introduction of such a notion through a resolution of the identity on the Cameron–Martin space associated with the abstract Wiener space is considered in this paper. The notion of vector fields adapted to this filtration and causal linear operators are introduced and geometric characterizations of these notions are derived. The notion of equivalence between two abstract Wiener spaces endowed with filtrations is defined. It is shown that under some general conditions, every abstract Wiener space endowed with a filtration is equivalent to a classical Wiener space of a finite or countably infinite collection of independent one-dimensional Wiener processes. Letube an adapted vector field andδuthe divergence ofu; some sufficient conditions for the exponential integrability ofδuare derived
The Hardy Class of Geometric Models and the Essential Spectral Radius of Composition Operators

AbstractLetφbe an analytic map of the disk into itself (that fixes the origin). Then, it is well known thatφinduces a bounded composition operatorCφon the Hardy spaceH2(D). Also, by a classical result of Kœnigs there is an analytic mapσsuch thatσ∘φ=λσ(λ=φ′(0)≠0), which is one-to-one whenφis one-to-one. Recently, P. Bourdon and J. Shapiro found a lower bound for the essential spectral radius ofCφin terms ofλandh(σ)=sup{p>0:σ∈Hp(D)}, and asked for an exact formula. In the following, we assume thatφis one-to-one. We determine the Hardy class ofσin terms of the image setG=σ(D), and then show that Bourdon and Shapiro's lower bound is also an upper bound, hence answering their question in the case of univalent symbols. Moreover, in an earlier paper, Shapiro, Smith, and Stegenga introduced the concept of twisted sectors to estimateh(σ), and we showed [P. Poggi-Corradini,Trans. Amer. Math. Soc.348(6) (1996), 2503–2518] how the dynamics of the mapφand some harmonic measure estimates allow one to completely characterize the caseh(σ)=∞ in terms of twisted sectors. By further analyzing this concept, we produce a new characterization ofh(σ) in terms of the invariant sector-like regions that are contained inG=σ(D). This different approach also yields another proof of Bourdon and Shapiro's lower bound mentioned above, in our specific case
Fine Structure of Zoll Spectra

AbstractThis paper is concerned with the wave equation on a Zoll manifold, i.e. a compact Riemannian manifold (M, g) all of whose geodesics are closed. By a result of V. Guillemin, such metrics exist in abundance onS2: for any odd functionfonS2there exists a one-parameter deformation of the canonical metricgothru Zoll metrics,gt=eftwithft=tf+…. And, as has been known for a long time [DG][CV.1][HoIV][W.1,2], the periodicity of the geodesic flow leads to a near periodicity in the wave groupUt=eitΔ. Precisely, there exists a unitary equivalenceΔ=R+Q−1where [R, Q−1]=0, where the spectrum ofRlies on an arithmetic progression and whereQ−1is a pseudodifferential operator of order −1. Hence the eigenvalues fall into clusterCkaround spec(R) with widthsO(k−1). The main result of this paper is to determine the limit distribution of the eigenvalues in the clusters. It was long ago shown by Weinsten [loc. cit] that (when properly normalized) the limit distribution has the formσQ−1dμwheredμis the Liouville measure onS*M. We complete the calculation by determiningσQ−1. It was conjectured [W.2][Ku] that up to universal constants,σQ−1=R(τ) whereRis the Radon transform of (M, g) andτis the scalar curvature [W.2][Ku]; but it turns out there is a second term involving the Jacobi fields. In addition, we show that the cluster projections define almost isometric minimal immersions of (M, g) into spheres and that in the case of maximally degenerate Zoll metrics the errors are of rapid decay
On Weak Compactness in Spaces of Measures

AbstractIt is proved that a weak* compact subsetAof scalar measures on aσ-algebra is weakly compact if and only if there exists a nonnegative scalar measureλsuch that each measure inAisλ-continuous (such a measureλis called a control measure forA). This result is then used to obtain a very general form of the Vitali–Hahn–Saks Theorem on finitely additive vector measures. Finally, it is proved that a weak* compact subsetAof regular Borel measures on anF-space is weakly compact if and only if there exists a nonnegative regular Borel measureλsuch that each measure inAisλ-continuous. This latter result shows that Grothendieck's theorem on weak* convergent sequences of measures is valid not only for weak* convergent sequences but also for weak* compact subsets with a control measure
Wavelets and the Angle between Past and Future

AbstractThe main result, the Riesz projectionP+(or, equivalently, Hilbert TransformT), is bounded in the weighted spaceL2(W) whereWis a matrix-valued weight if and only ifsupi‖[1|I|∫IW]1/2[1|I|∫IW−1]1/2‖<∞wheresupremumis taken over all intervalsI. Motivation for this problem comes from stationary processes (Riesz projection is bounded means the angle between “past” and “future” of a stationary process with spectral measureWis nonzero). In the scalar case the result is the well known Hunt–Muckenhoupt–Wheeden theorem. The main step in our proof is to show that a vector Haar system forms an unconditional basis inL2(W). As a byproduct of our approach we get some new results about bases of wavelets in weighted spaces (in both scalar and vector-valued cases)
Regular Level Sets of Averages of Nemytskiı Operators Are Contractible

AbstractLetH1(S1) be the space of periodic real functions with derivative inL2andf:R→R be a smooth function with no double roots. Then there is a diffeomorphism ofH1(S1) taking the setZ={v∈H1(S1)∣∫S1f(v(t))dt=0} to a hyperplane. In this paper we state and prove a general version of this example. We consider a Banach spaceVof functions from some manifoldMtoRn′and a functionf:M×Rn′→Rn: under suitable hypothesis, there is a homeomorphism ofVtakingZ={v∈V∣∫Mf(m, v(m))dm=0} to a closed subspace of codimensionn
Analytic Functions, Cauchy Formula, and Stationary Phase on a Real Abstract Wiener Space

AbstractA new complexification of a real abstract Wiener space will be introduced, and some analogs of the algebra of analytic functions on finite dimensional Euclidean space will be considered; analytic functions on the original space, their holomorphic prolongation to the complexified space, and holomorphic functions and a Cauchy formula on the complexified space. The Cauchy formula is a key tool to study probabilistic quantities via “deformation of the contour integration.” Namely, it will be applied to establish (i) an explicit representation of stochastic oscillatory integrals with quadratic phase function and (ii) a stationary phase estimation of the integrals. Further, the later estimation is applicable to study Gevrey type smoothness of density functions. An integration by parts formula on a totally real submanifold in the complexified space is also studied
Segal–Bargmann Transform, Functional Calculus on Matrix Spaces and the Theory of Semi-circular and Circular Systems

AbstractLetMd(C) be the space ofd×dcomplex matrices, and Xdbe the subspace of hermitian matrices. We study the Segal–Bargmann transform of functions on Xd, with values inMd(C), which are given by functional calculus. We show that whend→∞, the transform of such a map becomes close to the space of holomorphic functional calculus onMd(C), and that this yields, in the limit, an isometry between theL2space of Wigner's semi-circle distribution and the Hardy space of the disk. We relate this isometry to Voiculescu's theory of circular and semi-circular systems, and we study its analogue when the Segal–Bargmann transform is replaced by the Hall transform on unitary groups of large dimensions
Bergman Projections and Operators on Hardy Spaces

AbstractWe show that “Toeplitz like” operators of the formTsuf=Ps(uf), wherePsis a weighted Bergman projection, are bounded on the Hardy spacesHp, for 1⩽p<∞ for certain “symbols”udefined on the unit disk. In particular,Tsuis bounded ifuis of the formu=h+Gμwherehis a bounded harmonic function andGμis the Green potential of a Borel measureμsatisfying the condition that (1−|z|)dμ(z) is a Carleson measure. Ifuis of the formu=h+Gμwhereμis a positive measure then these two conditions are also necessary thatTsube bounded onHp. We also consider the cases 0<p<1
Completely Positive Maps of the Cuntz Algebras

AbstractWe construct a covariant functor from the category whose objects are the complex, infinite dimensional, separable Hilbert spaces and whose morphisms are the contractions into the category whose objects are the unitalC*-algebras and whose morphisms are the completely positive, identity-preserving maps. TheC*-algebras involved are isomorphic to the Cuntz algebra O∞and our functor extends the canonical action of the group of unitaries on the generating Hilbert space. Our result cannot apply to On,n<∞
Singular Inner Functions and Biinvariant Subspaces for Dissymetric Weighted Shifts

AbstractWe consider dissymetric weights on Z , i.e., nonincreasing unbounded weightsω:Z→[1, ∞) such thatω(n)=1 forn⩾0 and such that limn→−∞ω(n)1/|n|=1 whereω(n)=supp∈Zω(n+p)/ω(p). A dissymetric weight is said to be quasianalytic if ∑n<0logω(n)/n2=+∞. Denote byΓthe unit circle and byDthe open unit disc. SetL2ω(Γ)={f∈L2(Γ)∣‖f‖ω=(∑n∈Z|f(n)|2·ω2(n))1/2<+∞}.We identify the usual Hardy spaceH2(D) to the set {f∈L2ω(Γ)∣f(n)=0 (n<0)} and we denote bySω:f(eit)→eit·f(eit) the usual shift onL2ω(Γ). IfUis a singular inner function denote byEUthe closure of Span(Snω·U)n∈ZinL2ω(Γ). We show that ifωis a dissymetric weight such thatlogω(−n)n→n→∞∞thenEUis a proper subspace ofL2ω(Γ) for every singular inner functionU. Also ifωis any dissymetric weight then there exists some singular inner functionUsuch thatEUis a proper subspace ofL2ω, so thatSωalways has nontrivial biinvariant subspaces (no nontrivial biinvariant subspaces ofSωwere known so far in the quasianalytic case)
Some Sufficient Conditions for the Division Property of Invariant Subspaces in Weighted Bergman Spaces

AbstractA weighted Bergman spaceBis a Banach space of the formLp(μ)∩Hol(Ω), whereμis a Borel measure carried by the bounded regionΩin the complex plane. We consider closed subspacesMofBthat are invariant for multiplication by the independent variablez. We sayMhas the division property, if dimM/(z−λ)M=1 for eachλ∈Ω. In terms of the local boundary behavior of the functions inMwe give several conditions which imply the division property. For example, this happens ifMis generated by functions that extend analytically near a fixed boundary point and if ∂Ωis nice near this point. “Analytic” may be replaced by “locally Nevanlinna.” For the standard weights (1−|z|)αdAon the unit disc we show thatMhas the division property if it contains one function that is locally Nevanlinna near a boundary point. Furthermore, in the unweighted case (α=0) the invariant subspace generated by two functions that areLrrespectivelyLs(1/r+1/s=1/p) near some boundary point, has the division property
Graphs, Groupoids, and Cuntz–Krieger Algebras

AbstractWe associate to each locally finite directed graphGtwo locally compact groupoidsGandG(★). The unit space ofGis the space of one–sided infinite paths inG, andG(★) is the reduction ofGto the space of paths emanating from a distinguished vertex ★. We show that under certain conditions theirC*-algebras are Morita equivalent; the groupoidC*-algebraC*(G) is the Cuntz–Krieger algebra of an infinite {0, 1} matrix defined byG, and that the algebrasC*(G(★)) contain theC*-algebras used by Doplicher and Roberts in their duality theory for compact groups. We then analyse the ideal structure of these groupoidC*-algebras using the general theory of Renault and calculate theirK-theory
Ergodicity ofL2-Semigroups and Extremality of Gibbs States

AbstractWe extend the classical results of Holley–Stroock on the characterization of extreme Gibbs states for the Ising model in terms of the irreducibility (resp. ergodicity) of the corresponding Glauber dynamics to the case of lattice systems with unbounded (linear) spin spaces. We first develop a general framework to discuss questions of this type using classical Dirichlet forms on infinite dimensional state spaces and their associated diffusions. We then describe concrete applications to lattice models with polynomial interactions (i.e., the discreteP(ϕ)d-models of Euclidean quantum field theory). In addition, we prove the equivalence of extremality and shift-ergodicity for tempered Gibbs states of these models and also discuss this question in the general framework
Estimates of Logarithmic Sobolev Constant: An Improvement of Bakry–Emery Criterion

AbstractThis paper is mainly devoted to estimate the logarithmic Sobolev (abbrev. L.S.) constant for diffusion operators on manifold or in Rd. In most cases, we study the lower bounds but a generalization to [A. Korzeniowski,J. Funct. Anal.71(1987), 363–370, Theorem 1] for the upper bound is also presented (Theorem 1.5). Based on a simple observation (due to [J.-D. Deuschel and D. W. Stroock,J. Funct. Anal.92(1990), 30–48]) of the comparison between the L.S. constants for different potentials, the powerful Bakry–Emery criterion for the L.S. inequality is improved considerably in the paper, especially for the manifolds with non-positive sectional curvatures (Theorem 1.3(1)). In terms of our notation:β(r)=infρ(x, p)⩾rinfX∈Tx(M), ‖X‖=1(Ricc−HessV)(X, X), whereρ(x, p) is the distance betweenxand an arbitrary fixed pointp∈M, the improvement can be roughly stated as follows. The condition “infr⩾0β(r)>0” for which the criterion is available is now replaced by “supr⩾0β(r)>0.
Spectral Sets and Factorizations of Finite Abelian Groups

AbstractAspectral setis a subsetΩofRnwith Lebesgue measure 0<μ(Ω)<∞ such that there exists a setΛof exponential functions which form an orthogonal basis ofL2(Ω). The spectral set conjecture of B. Fuglede states that a set 0 is a spectral set if and only ifΩtilesRnby translation. We study setsΩwhich tileRnusing a rational periodic tile set S=Zn+A, where A⊆(1/N1)Z×…×(1/Nn)Zis finite. We characterize geometrically bounded measurable setsΩthat tileRnwith such a tile set. Certain tile sets S have the property that every bounded measurable setΩwhich tilesRnwith S is a spectral set, with a fixed spectrumΛS. We callΛSa universal spectrum for such S. We give a necessary and sufficient condition for a rational periodic setΛto be a universal spectrum for S, which is expressed in terms of factorizationsA⊕B=GwhereG=ZN1×…×ZNn, andA:=A (modZn). In dimensionn=1 we show that S has a universal spectrum wheneverN1is the order of a “good” group in the sense of Hajós, and for various other sets S
Non-homeomorphic Nilmanifolds with Identical Unitary Spectrum

AbstractWe show in every three-step nilpotent Lie groupGwith all coadjoint orbits flat the existence of a pair of discrete cocompact subgroupsΓ1andΓ2such thatΓ1\GandΓ2\Ghave the same unitary spectrum butΓ1is not isomorphic toΓ2. This result generalizes an example of Gornet. We mention without giving a proof a result that would enunciate that for a large category of subgroups of the four dimensional three-step chain group with non-flat coadjoint orbits, this phenomenon of non-isomorphic representation equivalence cannot occur. We also prove some short structural results for three-step nilpotent Lie groups with one-dimensional center
An Equivariant Brauer Group and Actions of Groups onC*-Algebras

AbstractSuppose that (G, T) is a second countable locally compact transformation group given by a homomorphism ℓ:G→Homeo(T), and thatAis a separable continuous-traceC*-algebra with spectrumT. An actionα:G→Aut(A) is said to cover ℓ if the induced action ofGonTcoincides with the original one. We prove that the set BrG(T) of Morita equivalence classes of such systems forms a group with multiplication given by the balanced tensor product: [A, α][B, β]=[A⊗C0(T)B, α⊗β], and we refer to BrG(T) as the Equivariant Brauer Group. We give a detailed analysis of the structure of BrG(T) in terms of the Moore cohomology of the groupGand the integral cohomology of the spaceT. Using this, we can characterize the stable continuous-traceC*-algebras with spectrumTwhich admit actions covering ℓ. In particular, we prove that ifG=R, then every stable continuous-traceC*-algebra admits an (essentially unique) action covering ℓ, thereby substantially improving results of Raeburn and Rosenberg
Renorming Banach Spaces with the Mazur Intersection Property

AbstractIn this paper we give new sufficient and necessary conditions for a Banach space to be equivalently renormed with the Mazur intersection property. As a consequence, several examples and applications of these results are obtained. Among them, it is proved that every Banach space embeds isometrically into a Banach space with the Mazur intersection property, answering a question asked by Giles, Gregory, and Sims. We also prove that for every treeT, the spaceC0(T) admits a norm with the Mazur intersection property, solving a problem posed by Deville, Godefroy, and Zizler. Finally, assuming the continuum hypothesis, we find an example of an Asplund space admitting neither an equivalent norm with the above property nor a nicely smooth norm