pinecone/core-2020-05-10-deduplication huggingface dataset:
Product Rule and Chain Rule Estimates for Fractional Derivatives on Spaces that Satisfy the Doubling Condition

AbstractThe purpose of this paper is to prove some classical estimates for fractional derivatives of functions defined on a Coifman–Weiss space of homogeneous type, in particular the product rule and chain rule estimates in (T. Kato and G. Ponce, 1988, Comm. Pure Appl. Math.41, 891–907) and (F. M. Christ and M. I. Weinstein, 1991, J. Funct. Anal.100, 87–109). The fractional calculus of M. Riesz was extended to these spaces in (A. E. Gatto, C. Segovia, and S. Vági, 1996, Rev. Mat. Iberoamericana12). Our main tools are fractional difference quotients and the square fractional derivative of R. Strichartz in (1967, J. Math. Mech.16, 9) extended to this context. For the particular case of Rn, our approach unifies the proofs of these estimates and clarifies the role of Calderón's formula for these results. Since the square fractional derivative can be easily discretized, we also show that the discrete and continuous Triebel–Lizorkin norms for fractional Sobolev spaces on spaces of homogeneous type are equivalent
Rank One Perturbations in a Pontryagin Space with One Negative Square

AbstractLet N1 denote the class of generalized Nevanlinna functions with one negative square and let N1, 0 be the subclass of functions Q(z)∈N1 with the additional properties limy→∞Q(iy)/y=0 and limsupy→∞y|ImQ(iy)|<∞. These classes form an analytic framework for studying (generalized) rank one perturbations A(τ)=A+τ[·, ω]ω in a Pontryagin space setting. Many functions appearing in quantum mechanical models of point interactions either belong to the subclass N1, 0 or can be associated with the corresponding generalized Friedrichs extension. In this paper a spectral theoretical analysis of the perturbations A(τ) and the associated Friedrichs extension is carried out. Many results, such as the explicit characterizations for the critical eigenvalues of the perturbations A(τ), are based on a recent factorization result for generalized Nevanlinna functions
Matrix Valued Spherical Functions Associated to the Complex Projective Plane

AbstractThe main purpose of this paper is to compute all irreducible spherical functions on G=SU(3) of arbitrary type δ∈K, where K=S(U(2)×U(1))≃U(2). This is accomplished by associating to a spherical function Φ on G a matrix valued function H on the complex projective plane P2(C)=G/K. It is well known that there is a fruitful connection between the hypergeometric function of Euler and Gauss and the spherical functions of trivial type associated to a rank one symmetric pair (G, K). But the relation of spherical functions of types of dimension bigger than one with classical analysis has not been worked out even in the case of an example of a rank one pair. The entries of H can be described by solutions of two systems of ordinary differential equations. There is no ready-made approach to such a pair of systems or even to a single system of this kind. In our case the situation is very favorable and the solution to this pair of systems can be exhibited explicitly in terms of a special class of generalized hypergeometric functions p+1Fp
Approximating Spectral Invariants of Harper Operators on Graphs

AbstractWe study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada (Sun). A main result in this paper is that the spectral density function of DMLs associated to rational weight functions on graphs with a free action of an amenable discrete group can be approximated by the average spectral density function of the DMLs on a regular exhaustion, with either Dirichlet or Neumann boundary conditions. This then gives a criterion for the existence of gaps in the spectrum of the DML, as well as other interesting spectral properties of such DMLs. The technique used incorporates some results of algebraic number theory
Global Existence for a Quasilinear Wave Equation Outside of Star-Shaped Domains

AbstractWe prove global existence of small-amplitude solutions of quasilinear Dirichlet-wave equations outside of star-shaped obstacles in (3+1)-dimensions. We use a variation of the conformal method of Christodoulou. Since the image of the space-time obstacle is not static in the Einstein diamond, our results do not follows directly from local existence theory as did Christodoulou's for the nonobstacle case. Instead, we develop weighted estimates that are adapted to the geometry. Using them and the energy-integral method we obtain solutions in the Einstein diamond minus the dime-dependent obstacle, which pull back to solutions in Minkowski space minus and obstacle
Strong Stability Results for Solutions of Elliptic Equations with Power-like Lower Order Terms and Measure Data

AbstractLet un be the sequence of solutions of−div(a(x, un, ∇un))+|un|q−1un=fninΩ,un=0on∂Ω, where Ω is a bounded set in RN and fn is a sequence of functions which is strongly convergent to a function f in L1loc(Ω\K), with K a compact in Ω of zero r-capacity; no assumptions are made on the sequence fn on the set K. We prove that if a has growth of order p−1 with respect to ∇u(p>1), and if q>r(p−1)/(r−p), then un converges to u, the solution of the same problem with datum f, thus extending to the nonlinear case a well-known result by H. Brezis
Functional Calculus on BMO and Related Spaces

AbstractLet f be a Borel measurable function of the complex plane to itself. We consider the nonlinear operator Tf defined by Tf[g]=fg, when g belongs to a certain subspace X of the space BMO(Rn) of functions with bounded mean oscillation on the Euclidean space. In particular, we investigate the case in which X is the whole of BMO, the case in which X is the space VMO of functions with vanishing mean oscillation, and the case in which X is the closure in BMO of the smooth functions with compact support. We characterize those f's for which Tf maps X to itself, those f's for which Tf is continuous from X to itself, and those f's for which Tf is differentiable in X
An Interpolation Theorem with Perturbed Continuity

AbstractLet (A0, A1) and (B0, B1) be two interpolation couples and let T:(A0, A1)↦ (B0, B1) be a K-quasilinear operator. The boundedness of the operator from A0 to B0 implies K(t, Ta; B0, B1)⩽M0‖a‖A0 and the boundedness of the operator from A1 to B1 implies K(t, Ta;B1, B0)⩽M1‖a‖A1, a∈A0∩A1. We consider perturbations of these two inequalities in the formK( γt, Ta; B 0, B 1)⩽M0‖a‖ A0+ε 0K (t, Ta;B 0, B 1) andK(γt, Ta;B 1, B 0)⩽M1‖a‖ A1+ε1K (t, Ta;B1, B0). We prove that similar to the classical case for γ>1 and 0⩽εj⩽1 we get for all 0<θ<1 and for all 0<q⩽∞‖Ta‖(B0, B1)θ , q⩽C(M0, M1,  θ, q)‖a‖(A0, A1)θ, q. If we take B0=L1(Q), B1=L∞(Q), γ=2, and εj=1, where Q is a cube in Rn, we get a theorem of Bennett, DeVore, and Sharpley. We prove that if εj>1 we continue to get an interpolation theorem, but the interpolation holds for (log+ε0)/(logγ)<θ<1−(log+ε1)/(logγ). This is the first instance of an interpolation theorem which holds for a subinterval of 0<θ<1. Bennett, DeVore and Sharpley identified a “weak L∞” class as the rearrangement invariant span of BMO(Q). This prompts the natural question of the existence of abstract “weak type” classes near the endpoints of interpolation scales. As we note below, there have been previous attempts to develop the theory of such classes. The construction was, however, too rigid, and necessitated the precise identification of the K -functional for the interpolation couple. We define these classes here in a way which allows for the identification K-functionals up to multiplicative equivalence. This opens the door to applications of the theory to most interpolation couples, leading to stronger interpolation theorems even for some well known spaces
Some Extensions of Loewner's Theory of Monotone Operator Functions

AbstractSeveral extensions of Loewner's theory of monotone operator functions are given. These include a theorem on boundary interpolation for matrix-valued functions in the generalized Nevanlinna class. The theory of monotone operator functions is generalized from scalar-to matrix-valued functions of an operator argument. A notion of κ-monotonicity is introduced and characterized in terms of classical Nevanlinna functions with removable singularities on a real interval. Corresponding results for Stieltjes functions are presented
Approximately Finitely Acting Operator Algebras

AbstractLet E be an operator algebra on a Hilbert space with finite-dimensional C*-algebra C*(E). A classification is given of the locally finite algebras A0=[formula](Ak, φk) and the operator algebras A=[formula](Ak, φk) obtained as limits of direct sums of matrix algebras over E with respect to star-extendible homomorphisms. The invariants in the algebraic case consist of an additive semigroup, with scale, which is a right module for the semiring VE=Homu(E⊗K, E⊗K) of unitary equivalence classes of star-extendible homomorphisms. This semigroup is referred to as the dimension module invariant. In the operator algebra case the invariants consist of a metrized additive semigroup with scale and a contractive right module VE-action. Subcategories of algebras determined by restricted classes of embeddings, such as 1-decomposable embeddings between digraph algebras, are also classified in terms of simplified dimension modules
An Lp Analog to AAK Theory for p⩾2

AbstractWe develop an Lp analog to AAK theory on the unit circle that interpolates continuously between the case p=∞, which classically solves for best uniform meromorphic approximation, and the case p=2, which is equivalent to H2-best rational approximation. We apply the results to the uniqueness problem in rational approximation and to the asymptotic behaviour of poles of best meromorphic approximants to functions with two branch points. As pointed out by a referee, part of the theory extends to every p∈[1, ∞] when the definition of the Hankel operator is suitably generalized; this we discuss in connection with the recent manuscript by V. A. Prokhorov, submitted for publication
Central Intertwining Lifting, Suboptimization, and Interpolation in Several Variables

AbstractIn this paper we obtain an explicit central intertwining lifting for row contractions. This is used to prove Kaftal–Larson–Weiss and Foiaş–Frazho H∞−H2 suboptimization type results for the noncommutative (resp. commutative) analytic Toeplitz algebra F∞n (resp. W∞n). The algebra F∞n (resp. W∞n) can be viewed as a multivariable noncommutative (resp. commutative) analogue of the Hardy space H∞. Similar results are provided for F∞n⊗B(K, K′) and W∞n⊗B(K, K′), where B(K, K′) is the set of all bounded linear operators acting on Hilbert spaces. New extensions of the Sarason, Carathéodory, and Nevanlinna–Pick type interpolation results are obtained for F∞n⊗B(K, K′) and some consequences to the operator-valued analytic interpolation in the unit ball of Cn are considered
Singular Pseudodifferential Operators, Symmetrizers, and Oscillatory Multidimensional Shocks

AbstractWe introduce a calculus of singular pseudodifferential operators (SPOs) depending on wavelength ε and use them to solve three different types of singular quasilinear hyperbolic systems. Such systems arise in nonlinear geometric optics and also, for example, in the study of incompressible limits and of nonlinear wave equations with small nonlinear terms or small data. The SPOs act in both slow and fast variables and are singular not only because their symbols have finite regularity and depend on 1ε, but also because their derivatives fail to decay in the usual way in the dual variables. There is a necessarily crude calculus with large parameter (e.g., residual operators are just bounded on L2), but the calculus admits the proof of Garding inequalities and enables us to symmetrize and sometimes even diagonalize the singular systems being considered by microlocalizing simultaneously in both slow and fast variables. The paper culminates in a proof of the existence of oscillatory multidimensional shocks on a fixed time interval independent of the wavelength ε as ε→0. The use of SPOs allows us to eliminate the small divisor assumptions made in earlier work and also to construct more general oscillatory solutions in which elliptic boundary layers are present on one or both sides of the shock
A Generalization of Hankel Operators

AbstractWe introduce a class of operators, called λ-Hankel operators, as those that satisfy the operator equation S*X−XS=λX, where S is the unilateral forward shift and λ is a complex number. We investigate some of the properties of λ-Hankel operators and show that much of their behaviour is similar to that of the classical Hankel operators (0-Hankel operators). In particular, we show that positivity of λ-Hankel operators is equivalent to a generalized Hamburger moment problem. We show that certain linear spaces of noninvertible operators have the property that every compact subset of the complex plane containing zero is the spectrum of an operator in the space. This theorem generalizes a known result for Hankel operators and applies to λ-Hankel operators for certain λ. We also study some other operator equations involving S
Finite Energy Solutions of Maxwell's Equations and Constructive Hodge Decompositions on Nonsmooth Riemannian Manifolds

AbstractWe consider two basic potential theoretic problems in Riemannian manifolds: Hodge decompositions and Maxwell's equations. Here we are concerned with smoothness and integrability assumptions. In the context of Lp forms in Lipschitz domains, we show that both are well posed provided that 2−ε<p<2+ε, for some ε>0, depending on the domain. Our approach is constructive (in the sense that we produce integral representation formulas for the solutions) and emphasizes the intimate connections between the two problems at hand. Applications to other related PDEs, such as boundary problems for the Hodge Dirac operator, are also presented
Approximate Identities for Ideals of Segal Algebras on A Compact Group

AbstractWe show that every closed ideal of a Segal algebra on a compact group admits a central approximate identity which has the property, called condition (U), that the induced multiplication operators converge to the identity operator uniformly on compact sets of the ideal. This result extends a known one due to H. Reiter who has considered the problem under the condition that the Segal algebra is symmetric. We prove further that a closed right ideal of a Segal algebra on a compact group admits a left approximate identity satisfying condition (U) if and only if it is approximately complemented as a subspace of the Segal algebra; if in addition the Segal algebra is symmetric, then a closed left ideal admits a right approximate identity satisfying condition (U) if and only if it is approximately complemented
Bergman-Type Reproducing Kernels, Contractive Divisors, and Dilations

AbstractLet Ω be a region in the complex plane. In this paper we introduce a class of sesquianalytic reproducing kernels on Ω that we call B-kernels. When Ω is the open unit disk D and certain natural additional hypotheses are added we call such kernels k Bergman-type kernels. In this case the associated reproducing kernel Hilbert space H(k) shares certain properties with the classical Bergman space L2α of the unit disk. For example, the weighted Bergman kernels kβw(z)=(1−wz)−β, 1⩽β⩽2 are Bergman-type kernels. Furthermore, for any Bergman-type kernel k one has H2⊆H(k)⊆L2a, where the inclusion maps are contractive, and Mζ, the operator of multiplication with the identity function ζ, defines a contraction operator on H(k). Our main results about Bergman-type kernels k are the following two: First, once properly normalized, the reproducing kernel for any nontrivial zero based invariant subspace M of H(k) is a Bergman-type kernel as well. For the weighted Bergman kernels kβ this result even holds for all Mζ-invariant subspace M of index 1, i.e., whenever the dimension of M/ζM is one. Second, if M is any multiplier invariant subspace of H(k), and if we set C∗=M⊖zM, then Mζ∣M is unitarily equivalent to Mζ acting on a space of C∗-valued analytic functions with an operator-valued reproducing kernel of the typekw(z)=(IC∗−zwV(z)V(w)*)kw(z), where V is a contractive analytic function V:D→L(E, C∗), for some auxiliary Hilbert space E. Parts of these theorems hold in more generality. Corollaries include contractive divisor, wandering subspace, and dilation theorems for all Bergman-type reproducing kernel Hilbert spaces. When restricted to index one invariant subspaces of H(kβ), 1⩽β⩽2, our approach yields new proofs of the contractive divisor property, the strong contractive divisor property, and the wandering subspace theorems and inner–outer factorization. Our proofs are based on the properties of reproducing kernels, and they do not involve the use of biharmonic Green functions as had some of the earlier proofs
Strong Hypercontractivity and Relative Subharmonicity

AbstractA Hermitian metric, g, on a complex manifold, M, together with a smooth probability measure, μ, on M determine minimal and maximal Dirichlet forms, QD and Qmax, given by Q(f)=∫Mg(gradf(z), gradf(z))dμ(z). QD is the form closure of Q on C∞c(M) and Qmax is the form closure of Q on C1b(M). The corresponding operators, AD and Amax, generate semigroups having standard hypercontractivity properties in the scale of Lp spaces, p>1, when the corresponding form, Q, satisfies a logarithmic Sobolev inequality. It was shown by the author (1999, Acta Math.182, 159–206) that the semigroup e−tAD has even stronger hypercontractivity properties when restricted to certain holomorphic subspaces of Lp. These results are extended here to Amax. When (M, g) is not complete it is necessary that the elliptic differential operator Amax degenerate on the boundary of M. A second proof of these strong hypercontractive inequalities for both AD and Amax is given, which depends on an extension of the submean value property of subharmonic functions. The Riemann surface for z1/n and the weighted Bergman spaces in the unit disc are given as examples
Representations in L2-Spaces on Infinite-Dimensional Symmetric Cones

AbstractIn this paper we study representations of the automorphism groups of classical infinite-dimensional tube domains. In particular we construct the L2-realization of all unitary highest weight representations, including the vector-valued case. We also find a projective representation of the full identity component of the affine automorphism group of the Hilbert–Schmidt version of the tube domain with trivial cocycle on the subgroup corresponding to the trace class version, but non-trivial on the large group. Finally we show that the operator-valued measures corresponding to the vector valued highest weight representations have densities of a rather weak type with respect to Wishart distributions which makes it possible to determine their “supports.
Generalized Eigenfunctions for Waves in Inhomogeneous Media

AbstractMany wave propagation phenomena in classical physics are governed by equations that can be recast in Schrödinger form. In this approach the classical wave equation (e.g., Maxwell's equations, acoustic equation, elastic equation) is rewritten in Schrödinger form, leading to the study of the spectral theory of its classical wave operator, a self-adjoint, partial differential operator on a Hilbert space of vector-valued, square integrable functions. Physically interesting inhomogeneous media give rise to nonsmooth coefficients. We construct a generalized eigenfunction expansion for classical wave operators with nonsmooth coefficients. Our construction yields polynomially bounded generalized eigenfunctions, the set of generalized eigenvalues forming a subset of the operator's spectrum with full spectral measure
Double Operator Integrals

AbstractThis paper is concerned with perturbation formulae of the form∥f(a)−f(b)∥Lp(M,τ)⩽K∥a−b∥ Lp(M,τ) with K>0 being a constant depending on p and f only, where f is a real-valued Lipschitz function and a,b are self-adjoint τ-measurable operators affiliated with a semifinite von Neumann algebra (M,τ), such that the difference a−b belongs to Lp(M,τ), 1<p<∞. In order to treat the situation where the von Neumann algebra M is not necessarily hyperfinite, we first develop an integration theory with respect to finitely additive spectral measures in a Banach space. Applied to product measures this integration theory may be considered as an abstract version of the double operator integrals due to Birman and Solomyak. To describe the class of integrable functions we employ our recent study of multiplier theory in UMD-spaces. Our perturbation formulae extend those of Davies and Birman–Solomyak for the case when M is a hyperfinite I∞-factor (i.e., for the Schatten p-classes). We also discuss analogous perturbation results in the setting of symmetric operator spaces associated with (M,τ)
Optimizing Improved Hardy Inequalities

AbstractLet Ω be a bounded domain in RN, N⩾3, containing the origin. Motivated by a question of Brezis and Vázquez, we consider an Improved Hardy Inequality with best constant b, that we formally write as: −Δ⩾(N−22)21∣x∣2+bV(x). We first give necessary conditions on the potential V, under which the previous inequality can or cannot be further improved. We show that the best constant b is never achieved in H01(Ω), and in particular that the existence or not of further correction terms is not connected to the nonachievement of b in H01(Ω). Our analysis reveals that the original inequality can be repeatedly improved by adding on the right-hand side specific potentials. This leads to an infinite series expansion of Hardy's inequality. The series obtained is in some sense optimal. In establishing these results we derive various sharp improved Hardy–Sobolev Inequalities
Symmetry Results for Solutions of Semilinear Elliptic Equations with Convex Nonlinearities

AbstractIn this paper, we study the symmetry properties of the solutions of the semilinear elliptic problem {−Δu=f(x,u)in Ωu=g(x)on ∂Ω, where Ω is a bounded symmetric domain in RN, N⩾2, and f:Ω×R→R is a continuous function of class C1 in the second variable, g is continuous and f and g are somehow symmetric in x. Our main result is to show that all solutions of the above problem of index one are axially symmetric when Ω is an annulus or a ball, g≡0 and f is strictly convex in the second variable. To do this, we prove that the nonnegativity of the first eigenvalue of the linearized operator in the caps determined by the symmetry of Ω is a sufficient condition for the symmetry of the solution, when f is a convex function
Limiting Behavior of the Ginzburg–Landau Functional

AbstractWe continue our study of the functional Eε(u)≔∫U12∣∇u∣2+14ε2(1−∣u∣2)2dx, for u∈H1(U;R2), where U is a bounded, open subset of R2. Compactness results for the scaled Jacobian of uε are proved under the assumption that Eε(uε) is bounded uniformly by a function of ε. In addition, the Gamma limit of Eε(uε)/(ln ε)2 is shown to be E(v)≔12∥v∥22+∥∇×v∥M, where v is the limit of j(uε)/∣ln ε∣, j(uε)≔uε×Duε, and ∥·∥M is the total variation of a Radon measure. These results are applied to the Ginzburg–Landau functionalFε(u,A;hext)≔∫U12∣∇Au∣2+14ε2(1−∣u∣2)2+12∣∇×A−hext∣ dx, with external magnetic field hext≈H∣ln ε∣. The Gamma limit of Fε/(ln ε)2 is calculated to beF(v,a;H)≔12[∥v−a∥22+∥∇×v∥M+∥∇×a−H∥22], where v is as before, and a is the limit of Aε/∣ln ε∣
On the Lp-Theory of C0-Semigroups Associated with Second-Order Elliptic Operators, II

AbstractWe study positive C0-semigroups on Lp associated with second-order uniformly elliptic divergence-type operators with singular lower-order terms, subject to a wide class of boundary conditions. We obtain an interval (pmin, pmax) in the Lp-scale where these semigroups can be defined, including the case 2∉(pmin, pmax). We present an example showing that the result is optimal. We also show that the semigroups are analytic with angles of analyticity and spectra of the generators independent of p, for the whole range of p where the semigroups are defined