pinecone/core-2020-05-10-deduplication huggingface dataset:
Stability of BSDEs with Random Terminal Time and Homogenization of Semilinear Elliptic PDEs

AbstractIn this paper, we extend the probabilistic method for homogenization of semi-linear parabolic PDEs, developed by Buckdahn, Hu, and Peng to the case of elliptic PDEs. First, we give a stability result for BSDEs with random terminal time which are related to elliptic PDEs as shown in Peng (Stochastics Stochastics Rep.37(1991), 61–74). In the one dimensional case, we also partially relax the monotonicity assumption on the coefficient. Then, we use these stability results for BSDEs with random terminal time to study homogenization of systems of semilinear elliptic PDEs
Harmonic Analysis on Conelike Bodies and Holomorphic Functions on Tube Domains

AbstractIn this paper, we establish Bochner–Weil type theorems and integral formulas of Lévy–Khinchin type in the setting of locally compact commutative semigroups with involution. These results are used to prove some new holomorphic extension results. For a conelike involution semigroup S in a finite dimensional real space V, we establish integral characterizations for continuous positive definite and definitizable functions on S. In particular, we obtain holomorphic and Cauchy–Riemann extensions of these functions by means of an integral representation. We prove that ifz0is a point in the complexified vector space W:=V+iV and if a function is defined onz0+S then either positive definiteness or definitizability offextends holomorphy from a domainΩcontainingz0to a tube domain in W containing the sum of 0 and the convex cone spanned by S. We also obtain the holomorphic extension as a generalized Fourier–Laplace transform of some measure or a Lévy–Khinchin type integral representation
On the Inverse Spectral Problem for the Camassa–Holm Equation

AbstractA key basis for seeking periodic solutions of the Camassa–Holm equation is to understand the associated spectral problemy′=14y+λmy.The periodic spectrum can be recovered from the norming constants and the elements of the auxiliary spectrum. The potential can then be reconstructed from the periodic spectrum. A necessary and sufficient condition for exponential decrease of the widthsλ2n−λ2n−1for a sequence 0<λ1⩽λ2<… of single or double eigenvalues tending to infinity is the real analyticity ofm. The case of a purely simple spectrum is typical of 0>m∈C1(R)
Discontinuous Maps from Lipschitz Algebras

AbstractFor an infinite compact metric space (X, d),α∈(0, 1) andf∈C(X), letpα(f)=sup{|f(t)−f(s)|/d(t, s)α:t, s∈X, t≠s}. The set Lipα(X, d)={f∈C(X):pα(f)<∞} with the norm ‖f‖α=|f|X+pα(f) is a Banach function algebra under pointwise multiplication. The subset lipα(X, d)={f∈Lipα(X, d):|f(t)−f(s)|/d(t, s)α→0 asd(t, s)→0} is a closed subalgebra of Lipα(X, d). Both Lipα(X, d) and lipα(X, d) are called Lipschitz algebras. We solve the problem of existence of homomorphisms and derivations from lipα(X, d) which are discontinuous on every dense subalgebra. In order to achieve that, we use the existence ofx∈Xand a linear functionalλfrommα(x)={f∈lipα(X, d):f(x)=0} which is discontinuous on every dense subalgebra, but satisfies |λ(fg)|⩽pα(f)pα(g) for allf, g∈mα(x). We present two constructions of such functionals, one for the general case of lipα(X, d), and another one for the special case lipα([0, 1]). We relate the present results to the results concerning the eventual continuity of homomorphisms from Lipschitz algebras. We conclude that for anyβ∈[α, min{2α, 1}) there exist homomorphisms and derivations discontinuous on lipδ(X, d) in ‖·‖δnorm for allδ∈[α, β]. In our constructions, all maps are continuous forδ>β
Unitary Highest Weight Representations in Hilbert Spaces of Holomorphic Functions on Infinite Dimensional Domains

AbstractAutomorphism groups of symmetric domains in Hilbert spaces form a natural class of infinite dimensional Lie algebras and corresponding Banach Lie groups. We give a classification of the algebraic category of unitary highest weight modules for such Lie algebras and show that infinite dimensional versions of the Lie algebras so(2,n) have no unitary highest weight representations and thus do not meet the physical requirement of having positive energy. Highest weight modules correspond to unitary representations of global Banach Lie groups realized in Hilbert spaces of vector valued holomorphic functions on the relevant domains in Hilbert spaces. The construction of such holomorphic representations of certain Banach Lie groups, followed by the application of the general framework of Harish-Chandra type groups in an appropriate Banach setting, leads to the integration of the Lie algebra representation to a group representation. The extension of this theory to infinite dimensional settings is explored
Spinor Currents as Vector Particles

AbstractWe study a prototype for the problem of determining the irreducible factors of the spaces spanned by currents between homogeneous vector bundles. More specifically, letGdenote the conformal group of Minkowski space–timeM0, or more precisely,SU(2,2). It is shown that the positive-energy unitaryG-irreducible massive factors of the one-form bundle over the conformal compactification ofM0are bundle-equivalent to spaces spanned by currents of the spinor bundle. This has the interpretation that theZ-particle may be modelled by the representation ofGof lowestK-type (“LKT”) (3,emsp14;1/2,1/2), whereK=SU(1)×SU(2)×SU(2) and (m,j,j′) refers to the representation in which the lowest eigenvalue of theSU(1) generator ism, whilejandj′ are the spins of the left and rightSU(2) representations. In particular, this “Z” representation is bundle-equivalent to the representation ofGin a space of currents between neutrinos, of LKT (3/2,0,1/2), and antineutrinos, of LKT (3/2,1/2,0). TheW±-particle representation may similarly be modelled by the representation ofGof lowestK-type (4,0,0), the only other massive positive-energy unitary factor of the one-form bundle. In particular, theW−is equivalent to a space of currents between the electron, of LKT (5/2,0,1/2), and the antineutrino. These results are consistent with the possibility that all quasi-stable elementary particles may be modelled by similar subspaces of bundle products
Characterizing Yang–Mills Fields by Stochastic Parallel Transport

AbstractWe give a new stochastic approach to Yang–Mills fields on vector bundles by studying the derivative of the stochastic parallel transport in the vector bundle under a variation of connection. We establish martingale criterions for variations transversal to the gauge orbit and for Yang–Mills fields using variations induced by the flow of vector fields of gradient type on the base manifold. In dimension four we link the local Yang–Mills action to the quadratic variation of the derivative process defined by scalings of the driving Brownian motion in a coordinate chart
Conformal Extension of Massive Wave Functions

AbstractWe extend from themasslessto themassivecase the canonical conformal mapping of solutions of wave equations in two-dimensional Minkowski spaceM0into the universal coverMof the conformal compactificationMofM0. There is a continuous Poincaré-covariant map from the space of solutions of the Klein–Gordon equation inM0to the dual of the space of analytic vectors for the action of the conformal groupG≃SU(1, 1)×SU(1, 1) on massive fields inM. The extensions satisfy a corresponding wave equation inM. As in the case of massless equations, the extended solutions are periodic under the generatorζof the infinite cyclic center of the conformal groupGinM. The analysis involves the determination of the action of the chronometric temporal evolution, i.e., the one-parameter group of nonlinear transformations generated by the center of the maximal essentially-compact subgroup ofGon the Fourier transforms of wave functions in Minkowski space. The analytic continuation of this evolution to complex times of positive imaginary part is represented by a Hilbert–Schmidt operator whose kernel is determined in closed form. Fields of higher conformal weight associated with other discrete series representations ofGare also treated in this respect
Distributions of Localized Eigenvalues of Laplacians on Post Critically Finite Self-Similar Sets

AbstractIn this paper, we study distributions of eigenvalues corresponding to localized eigenfunctions of Laplacians on p.c.f. self-similar sets. Precisely, we divide the eigenvalue counting functionρ(x) of a Laplacian into two parts,ρW(x) andρF(x), whereρW(x) is the counting function of localized eigenvalues andρF(x) is the counting function of non-localized (global) eigenvalues. We study asymptotic behaviors ofρW(x) andρF(x) asx→∞. It is shown thatρW(x)≈xdS/2wheredSis the spectral exponent. On the other hand, for a class of Laplacians, including the standard Laplacian on the Sierpinski gasket,ρF(x)≈xκFfor someκF<dS/2. So localized eigenfunctions dominate global eigenfunctions in such cases
The Langlands Parameters of Subquotients of Certain Derived Functor Modules

AbstractLetGbe a noncompact, simple Lie group with finite center, letKbe a maximal compact subgroup, and let g0=k0⊕p0be the corresponding decomposition of the Lie algebra. Suppose rankG=rankK, and let t0be a compact Cartan subalgebra and b be a Borel subalgebra. LetAb(λ) be a derived functor module with infinitesimal characterλ+δwhich is nondominant with respect to a noncompact simple root. Suppose thatΛ=λ+2δ(p) isKdominant so that theKtype,τΛ, with highest weightΛoccurs with multiplicity one inAb(λ). We develop conditions on the roots of g under which a functor of cohomological induction maps a certain module of parabolic induction to another module of parabolic induction, extending a result due to Vogan. This allows us, in many cases, to determine the Langlands parameters of the subquotient ofAb(λ) containingτΛ, via a conjectured method of Knapp
Finitely-Generated Algebras of Smooth Functions, in One Dimension

AbstractWe characterise the closure inC∞(R,R) of the algebra generated by an arbitrary finite point-separating set ofC∞functions. The description is local, involving Taylor series. More precisely, a functionf∈C∞belongs to the closure of the algebra generated byψ1,…,ψras soon as it has the “right kind” of Taylor series at each pointasuch thatψ′1(a)=…=ψ′r(a)=0. The “right kind” is of the formq∘(T∞aψ1−ψ1(a),…,T∞aψr−ψr(a)), whereqis a power series inrvariables, andT∞aψidenotes the Taylor series ofψiabouta
The Poisson Boundary of Teichmüller Space

AbstractFor a Markov operator on Teichmüller space commuting with the action of the mapping class group we prove convergence of sample paths of the associated Markov chain in the Thurston compactification and show that the Poisson boundary of the Markov operator can be identified with the space of projective measured foliations. The approach consists in using the authors' results on the Poisson boundary of the mapping class group in combination with a discretization procedure based on a Harnack inequality for Markov operators on Teichmüller space
Nonuniform Multiresolution Analyses and Spectral Pairs

AbstractA generalization of the notion of multiresolution analysis, based on the theory of spectral pairs, is considered. In contrast to the standard setting, the associated subspaceV0ofL2(R) has, as an orthonormal basis, a collection of translates of the scaling functionφof the form {φ(x−λ)}λ∈ΛwhereΛ={0,r/N}+2Z,N⩾1 is an integer, andris an odd integer with 1⩽r⩽2N−1 such thatrandNare relatively prime andZis the set of all integers. Furthermore, the corresponding dilation factor is 2N, the case whereN=1 corresponding to the usual definition of a multiresolution analysis with dilation factor 2. A necessary and sufficient condition for the existence of associated wavelets, which is always satisfied whenN=1 or 2, is obtained and is shown to always hold if the Fourier transform ofφis a constant multiple of the characteristic function of a set
Weighted Subcoercive Operators on Lie Groups

AbstractLetUbe a continuous representation of a Lie groupGon a Banach space X anda1, …, ad′an algebraic basis of the Lie algebra g ofG, i.e., thea1, …, ad′together with their multi-commutators span g. LetAi=dU(ai) denote the infinitesimal generator of the continuous one-parameter groupt↦U(exp(−tai)) and setAα=Ai1…Ainwhereα=(i1, …, in) withij∈{1, …, d′}. We analyze properties ofmth order differential operatorsdU(C)=∑α; |α|⩽mcαAαwith coefficientscα∈C. IfLdenotes the left regular representation ofGinL2(G) thendL(C) satisfies a Gårding inequality onL2(G) if, and only if, the closure of eachdU(C) generates a holomorphic semigroupSon X, the action ofSzis determined by a smooth, representation independent, kernelKzwhich, together with its derivativesAαKz, satisfiesmth order Gaussian bounds and, ifUis unitary,Sis quasi-contractive in an open representation independent subsector of the sector of holomorphy. Alternatively,dL(C) satisfies a Gårding inequality onL2(G) if, and only if, the closure ofdL(C) generates a holomorphic, quasi-contractive, semigroup satisfying bounds ‖AiSt‖2→2⩽ct−1/meωtfor allt>0 andi∈{1, …, d′}. These results extend to operators for which the directionsa1, …, ad′are given different weights. The unweighted Gårding inequality is a stability condition on the principal part, i.e., the highest-order part, ofdL(C) but in the weighted case the condition is on the part ofdL(C) with the highest weighted order
Homotopy of a Pair of Approximately Commuting Unitaries in a Simple C*-Algebra

AbstractIt is shown that, for a class of unital C*-algebras including purely infinite simple C*-algebras, real rank zero simple AT algebras, and AF algebras, ifuandvare almost commuting unitaries whereuhas trivialK1-class,vhas full spectrum, and a certainK0-valued obstruction associated to the pairu,vis trivial, thenucan be deformed to 1 through a path of unitaries in the algebra almost commuting withv, and the length of the path can be estimated by a universal constant. This result is used to identify the obstruction with Loring's Bott element associated to the pairu,vand also to prove the more universal statement that if (ui,vi),i=1,2, are two pairs of almost commuting unitaries with [ui]1, [vi]1, and Bott (ui,vi) each independent ofi, then one pair can be deformed into the other along a path of pairs of almost commuting unitaries in the algebra, the length of the path being bounded by a universal constant
Optimal Sobolev Inequalities of Arbitrary Order on Compact Riemannian Manifolds

AbstractLet (M,g) be a smooth compact RiemannianN-manifold,N⩾2, letp∈(1,N) real, and letHp1(M) be the Sobolev space of orderpinvolving first derivatives of the functions. By the Sobolev embedding theorem,Hp1(M)⊂Lp*(M) wherep*=Np/(N−p). Classically, this leads to some Sobolev inequality (I1p), and then to some Sobolev inequality (Ipp) where each term in (I1p) is elevated to the powerp. Long-standing questions were to know if the optimal versions with respect to the first constant of (I1p) and (Ipp) do hold. Such questions received an affirmative answer by Hebey–Vaugon forp=2. We prove here that forp>2, andp2<N, the optimal version of (Ipp) is false if the scalar curvature ofgis positive somewhere. In particular, there exist manifolds for which the optimal versions of (I1p) are true, while the optimal versions of (Ipp) are false. Among other results, we prove also that the assumption on the sign of the scalar curvature is sharp by showing that for anyp∈(1,N), the optimal version of (Ipp) holds on flat tori
Orthogonal Measures on the Boundary of a Riemann Surface and Polynomial Hull of Compacts of Finite Length

AbstractLetμbe an orthogonal measure with compact support of finite length in Cn. We prove, under a very weak hypothesis of regularity on the support (Suppμ) ofμ, that this measure is characterized by its boundary values (in the weak sense of currents) of the current [T]∧ϕ, whereTis an analytic subset of dimension 1 of Cn\Suppμandϕis a holomorphic (1, 0)-form onT. This allows us to prove that the polynomial hullXof a compactumX⊂Cnof finite length with a weak regularity assumption is its union with an analytic subset of pure dimension 1 of Cn\X. We also prove that the measureμcan be decomposed into a sum of orthogonal measures will small support. We deduce that a continuous function onXis approximable by polynomials if and only if it islocallyapproximable
Inversion of Fractional Integrals Related to the Spherical Radon Transform

AbstractExplicit inversion formulas are obtained for the analytic family of fractional integrals (Tαf)(x)=γn,α∫Sn|xy|α−1f(y)dyon the unit sphere in Rn+1. Arbitrary complexαandn⩾2 are considered. In the easeα=0 the integralTαfcoincides with the spherical Radon transform. Forα>1(α≠1,3,5,…) such integrals are known as the Blaschke–Levy representations and arise in convex geometry, probability, and the Banach space theory. Forα=1,3,5,… the integralTαfis defined by continuity as the spherical convolution with the power–logarithmic kernel. Different inversion methods are discussed
Endpoint Inequalities for Spherical Multilinear Convolutions

AbstractWriteσ=(σ1,…,σn) for an element of the sphereΣn−1and letdσdenote Lebesgue measure onΣn−1. For functionsf1,…,fnonR, defineT(f1,…,fn(x)=∫Σn−1f1(x−σ1)…fn(x−σn)dσ,x∈RLetR=R(n) denote the closed convex hull inR2of the points (0,0), (1/n,1), ((n+1)/(n+2),1), ((n+1)/(n+3), 2/(n+3)), ((n−1)/(n+1),0). We show that ifn⩾3, then the inequality‖T(f1,…,fn)‖q⩽C‖f1‖p…‖fn‖pholds if and only if (1/p,1/q)∈R. Our results fill in the gap in the necessary and sufficient conditions whenn⩾3 in Oberlin's previous work. A negative result is given along with some positive results, whenn=2, thus narrowing the gap in the necessary and sufficient conditions in this case
On Conformal Deformations of Metrics onSn

AbstractOnSn, there is a naturally metric definednth order conformal invariant operatorPn. Associated with this operator is a so-calledQ-curvature quantity. When two metrics are pointwise conformally related, their associated operators, together with theirQ-curvatures, satisfy the natural differential equations. This paper is devoted to the question of which function can be aQ-curvature candidate. This is the so-calledprescribing Q-curvature problem. Our main result is that ifQis positive, nondegenerate and the naturally defined mapping associated withQhas nonzero degree, then our problem has a solution. This is the natural generalization of prescribing Gaussian curvature onS2intoSn
Rank One Subspaces of Bimodules over Maximal Abelian Selfadjoint Algebras

AbstractSpaces of operators that are left and right modules over maximal abelian selfadjoint algebras (masa bimodules for short) are natural generalizations of algebras with commutative subspace lattices. This paper is concerned with density properties of finite rank operators and of various classes of compact operators in such modules. It is shown that every finite rank operator of a norm closed masa bimodule M is in the trace norm closure of the rank one subspace of M. An important consequence is that the rank one subspace of a strongly reflexive masa bimodule (that is, one which is the reflexive hull of its rank one operators) is dense in the module in the weak operator topology. However, in contrast to the situation for algebras, it is shown that such density need not hold in the ultraweak topology. A new method of representing masa bimodules is introduced. This uses a novel concept of anω-topology. With the appropriate notion ofω-support, a correspondence is established between reflexive masa bimodules and theirω-supports. It is shown that, if a C2-closed masa bimodule contains a trace class operator then it must contain rank one operators; indeed, every such operator is in the C2-norm closure of the rank one subspace of the module. Consequently the weak closure of any masa bimodule of trace class operators is strongly reflexive. However, the trace norm closure of the rank one subspace need not contain all trace class operators of the module. Also, it is shown that there exists a CSL algebra which contains no trace class operators yet contains an operator belonging to Cpfor allp>1. From this it follows that a transitive bimodule spanned by the rank one operators it contains need not be dense in Cpfor 1⩽p<∞. As an application, it is shown that there exists a commutative subspace lattice L such that L is non-synthetic but every weakly closed algebra which contains a masa and has invariant lattice L coincides with Alg L
A Dichotomy for Linear Spaces of Toeplitz Operators

AbstractLetSbe a linear manifold of bounded Hilbert space operators. An operatorAbelongs to thereflexive closureofSifAfbelongs to the closure ofSffor each vectorfin the underlying Hilbert space. Two extreme possibilities are (1)Sisreflexivein the sense that refS=S, and (2)Sistransitivein the sense that refSincludes all bounded operators on the underlying space. We show that every linear space B of Toeplitz operators which is closed in the ultraweak operator topology is either transitive or reflexive. No intermediate behavior is possible. The full space of all Toeplitz operators is transitive, but if B is properly contained in this space and contains allanalyticToeplitz operators, then B must be reflexive. In particular, the space of Toeplitz operators whose matrices have zeros on a fixed superdiagonal is reflexive
Embedding Theorems for Star-Invariant Subspaces Generated by Smooth Inner Functions

AbstractGiven an inner functionθon the unit disk D, letKpθ≔Hp∩θzHpbe the corre- sponding star-invariant subspace of the Hardy spaceHp. We are concerned with embedding theorems of the formKpθ⊂Lq(μ), whereμis a measure on D, and some related norm inequalities. In particular, assuming thatθhas a certain smoothness property, we characterize theμ's such that everyf∈K1θsatisfies∫D|f|qdμ1/q⩽const∫∂D|f||θ′|α|dz|,where 1⩽q<∞ and 0<α<1. This result sheds some light on the nature of Carleson-type measures forKpθ
On the Unitary Systems Affiliated with Orthonormal Wavelet Theory inn-Dimensions

AbstractWe consider systems of unitary operators on the complex Hilbert spaceL2(Rn) of the form U≔UDA,Tv1,…,Tvn≔{DmTl1v1…Tlnvn:m,l1,…,ln∈Z}, whereDAis the unitary operator corresponding to dilation by ann×nreal invertible matrixAandTv1,…,Tvnare the unitary operators corresponding to translations by the vectors in a basis {v1,…,vn} for Rn. Orthonormal waveletsψare vectors inL2(Rn) which are complete wandering vectors for U in the sense that {Uψ:U∈U} is an orthonormal basis forL2(Rn). It has recently been established that wheneverAhas the property that all of its eigenvalues have absolute values strictly greater than one (the expansive case) then U has orthonormal wavelets. The purpose of this paper is to determine when two (n+1)-tuples of the form (DA,Tv1,…,Tvn) give rise to the “same wavelet theory.” In other words, when is there a unitary transformation of the underlying Hilbert space that transforms one of these unitary systems onto the other? We show, in particular, that two systems UDA,Tei, UDB,Tei, each corresponding to translation along the coordinate axes, are unitarily equivalent if and only if there is a matrixCwith integer entries and determinant ±1 such thatB=C−1AC. This means that different expansive dilation factors nearly always yield unitarily inequivalent wavelet theories. Along the way we establish necessary and sufficient conditions for an invertible realn×nmatrixAto have the property that the dilation unitary operatorDAis a bilateral shift of infinite multiplicity