pinecone/core-2020-05-10-deduplication huggingface dataset:
Gradients on Fractals

AbstractIn this paper we define and study a gradient on p.c.f. (post critically finite, or finitely ramified) fractals. We use Dirichlet (energy) form analysis developed for such fractals by Kigami. We consider both nondegenerate and degenerate harmonic structures (where a nonzero harmonic function can be identically zero on an open set). We show that the energy is equal to the integral of a certain seminorm of the gradient if the harmonic structure is weakly nondegenerate. This result was proved by Kusuoka in a different form. We show that for a C1-function on the Sierpiński gasket the gradient considered here and Kusuoka's gradient essentially coincide with a gradient considered by Kigami. The gradient at a junction point was studied by Strichartz in relation to the Taylor approximation on fractals. He also proved the existence of the gradient almost everywhere with respect to the Hausdorff (Bernoulli) measure for a function in the domain of the Laplacian. In this paper we obtain certain continuity properties of the gradient for a function in the domain of the Laplacian. As an appendix, we prove an estimate of the local energy of harmonic functions which was stated by Strichartz as a hypothesis
Stability of Stationary Transport Equations with Accretive Collision Operators

AbstractIn this paper we consider transport equations with accretive collision operators. We characterize when the equation has a unique solution and show that in this case the solution is stable under small perturbations of the collision operator and the initial value. In one case in which there is more than one solution we show how to make a special selection of a solution, which is then stable again under small perturbations of both the collision operator and the initial value. The results obtained here parallel those obtained earlier for the case where the collision operator is positive semidefinite
Bases Consisting of Rational Functions of Uniformly Bounded Degrees or More General Functions

AbstractWe prove in this paper the existence of a Schauder basis for C[0, 1] consisting of rational functions of uniformly bounded degrees. This solves an open question of some years standing concerning the possible existence of such bases. This result follows from a more general construction of bases on R and [0, 1]. We prove that the new bases are unconditional bases for Lp, 1<p<∞, and Besov spaces. On [0, 1], they are Schauder bases for C[0, 1] as well. The new bases are utilized for nonlinear approximation
Factorization of Tent Spaces and Hankel Operators

AbstractIt is shown that the factorization Tpq=Tp∞·T∞q for tent spaces proved by R. R. Coifman et al. (1985, J. Funct. Anal.62, 304–335) for p>q, q=2, holds true for all 0<p, q<∞. From this certain strong factorization theorems are derived for spaces Hps of fractional derivatives of Hp functions, and more general Triebel spaces. In particular, it is proved that Hps=Hp·BMOAs . Applications considered include characterizations of symbols of bounded Hankel operators Hφ: Hp→Hqs, complex interpolation of tent spaces, and Carleson measure theorems for derivatives of Hp functions
Canonical Quantization on a Doubly Connected Space and the Aharonov–Bohm Phase

AbstractWe consider the canonical quantization (Schrödinger representation) on a doubly connected space ΩR≡R2\{(x, y)∣x2+y2⩽R2} (R>0). We show that, when we employ 2-dimensional orthogonal coordinates Ox1x2, there are uncountably many different self-adjoint extensions pUj of pj≡−i∂/∂xj (j=1, 2), and none of the pairs {pj, qj′}j, j′=1, 2 (qj′≡xj′·) satisfies the Weyl relation. Then, we construct a new canonical pair of canonical momentum and position operators so that the pair can satisfy the Weyl relation by using the streamline coordinates. As its application, in the Weyl relation with respect to the pair of the mv-momentum and position operators by the above new canonical pair, we find the Aharonov–Bohm phase
Uniqueness of Translation Invariant Norms

AbstractLet A be a Banach function space and let M be a family of multipliers on A. We provide conditions on M so that the original topology of A is the only complete norm topology on A making all of the maps from M continuous. As a corollary we show that for a compact abelian group G, and a circle group T•for A=Lp(T), 1<p<∞, the Lp-norm is the only one that makes all translations continuous, while •for A=C(G), A=L∞(G), or A=L1(G) there are other norms with that property. For noncompact groups the situation is different—on the space L1(R) the L1-norm is the only one that makes a single nontrivial translation continuous
Duality and Twisted Sums of Banach Spaces

AbstractWe give a negative answer to the three-space problem for the Banach space properties to be complemented in a dual space and to be isomorphic to a dual space (solving a problem of Vogt [Lectures held in the Functional Analysis Seminar, Dusseldorf/Wuppertal, Jan–Feb. 1987] and another posed by Dı&#x0301;az et al. in [Bull. Polish Acad. Sci. Math.40 (1992), 221–224]). Precisely, we construct an exact sequence 0→ℓ2→D→W*→0 in which W* is a separable dual and D is not isomorphic to a dual space. We also show the existence of an exact sequence 0→Y→X→Z→0 where both Y and Z are dual spaces and X is not even complemented in its bidual. To do that we perform a study of the basic questions on duality from the point of view of exact sequences of Banach spaces
Lp-Uniqueness of Schrödinger Operators and the Capacitary Positive Improving Property

AbstractWe prove several Lp-uniqueness results for Schrödinger operators −L+V by means of the Feynman–Kac formula. Using the (m, p)-capacity theory for general Markov semigroups, we show that the associated Feynman–Kac semigroup is positive improving in the sense of (m, p)-capacity, improving the well known one in the sense of measure. Using that capacitary positive improving property and two new inequalities for generalized Ornstein–Uhlenbeck generators, we show the essential self-adjointness of the ground state diffusion generator Lφ=L+2Γ(φ, ·)/φ associated with two dimensional Euclidean quantum fields
Free Composition of Paragroups

AbstractAs a first step we define an extremality condition for paragroups, then we prove that any given extremal paragroup is equivalent to a Popa system constructed with the string algebras associated with the starting paragroup. As a corollary, we get that any extremal paragroup can be thought of as the paragroup coming from a (generally non-hyperfinite) extremal inclusion of II1 factors. Then we define a free composition for paragroups, using as a model the definition by V. Jones and D. Bisch of free composition of subfactors; and we prove that for any two given paragroups P1 and P2 there always exists a third paragroup P which realizes the free composition of P1 and P2
Simple Purely Infinite C*-Algebras and n-Filling Actions

AbstractLet n be a positive integer. We introduce a concept, which we call the n-filling property, for an action of a group on a separable unital C*-algebra A. If A=C(Ω) is a commutative unital C*-algebra and the action is induced by a group of homeomorphisms of Ω then the n-filling property reduces to a weak version of hyperbolicity. The n-filling property is used to prove that certain crossed product C*-algebras are purely infinite and simple. A variety of group actions on boundaries of symmetric spaces and buildings have the n-filling property. An explicit example is the action of Γ=SLn(Z) on the projective n-space
Best Constants for the Riesz Projection

AbstractWe prove the following inequality with a sharp constant,‖P+f‖L p(T)⩽csc πp ‖f‖Lp(T),f∈Lp(T),where 1<p<∞, and P+: Lp(T)→Hp(T) is the Riesz projection onto the Hardy space Hp(T) on the unit circle T. (In other words, the “angle” between the analytic and co-analytic subspaces of Lp(T) equals π/p* where p*=max (p, pp−1).) This was conjectured in 1968 by I. Gohberg and N. Krupnik. We also prove an analogous inequality in the nonperiodic case where P+f=F−1 (χR+Ff) is the half-line Fourier multiplier on R. Similar weighted inequalities with sharp constants for Lp(R, |x|α), −1<α<p−1, are obtained. In the multidimensional case, our results give the norm of the half-space Fourier multiplier on R
Operator Monotone Functions which Are Defined Implicitly and Operator Inequalities

AbstractThe function tα (0<α<1) is operator monotone on 0⩽t<∞. This is known as the Löwner–Heinz inequality. However, not too many examples of concrete operator monotone functions are known so far. We will systematically seek operator monotone functions which are defined implicitly. This investigation is new, and our method seems to be powerful. We will actually find a family of operator monotone functions which includes tα (0<α<1). Moreover, by constructing one-parameter families of operator monotone functions, we will get many operator inequalities; especially, we will extend the Furuta inequality and the exponential inequality of Ando
Potential Theory on Lipschitz Domains in Riemannian Manifolds: Sobolev–Besov Space Results and the Poisson Problem

AbstractWe continue a program to develop layer potential techniques for PDE on Lipschitz domains in Riemannian manifolds. Building on Lp and Hardy space estimates established in previous papers, here we establish Sobolev and Besov space estimates on solutions to the Dirichlet and Neumann problems for the Laplace operator plus a potential, on a Lipschitz domain in a Riemannian manifold with a metric tensor smooth of class C1+γ, for some γ>0. We treat the inhomogeneous problem and extend it to the setting of manifolds results obtained for the constant-coefficient Laplace operator on a Lipschitz domain in Euclidean space, with the Dirichlet boundary condition, by D. Jerison and C. Kenig
Finite Dimensional de Branges Spaces on Riemann Surfaces

AbstractWe study certain finite dimensional reproducing kernel indefinite inner product spaces of multiplicative half order differentials on a compact real Riemann surface; these spaces are analogues of the spaces introduced by L. de Branges when the Riemann sphere is replaced by a compact real Riemann surface of a higher genus. In de Branges theory an important role is played by resolvent-like difference quotient operators Rα; here we introduce generalized difference quotient operators Ryα for any non-constant meromorphic function y on the Riemann surface. The spaces we study are invariant under generalized difference quotient operators and can be characterized as finite dimensional indefinite inner product spaces invariant under two operators Ry1αi and Ry2α2, where y1 and y2 generate the field of meromorphic functions on the Riemann surface, which satisfy a supplementary identity, analogous to the de Branges identity for difference quotients. Just as the classical de Branges spaces and difference quotient operators appear in the operator model theory for a single nonselfadjoint (or nonunitary) operator, the spaces we consider and generalized difference quotient operators appear in the model theory for commuting nonselfadjoint operators with finite nonhermitian ranks
Embeddings of Weakly Compact Sets and *-Paired Banach Spaces

AbstractWe use properties of Day's norm on c0(κ) to prove that, for every Eberlein compact space K, there exists a separately continuous symmetric mapping d:K×K→R such that we haved(x, y)<d(x, x)+d(y, y)2for any two distinct points x and y of K. As a consequence, we have that every Eberlein compact space K can be embedded as a point-separating set in its own function space C(K) equipped with pointwise (or weak) topology; in the terminology of Arkhangel'skii, this means that every Eberlein compact space is “self-dual.” We consider whether every Eberlein compact space K can be embedded as a generating set in C(K) (equipped with the weak topology). We show that such an embedding exists for every uniformly Eberlein compact space. We also show that every Eberlein compact space can be embedded as a free generating set in some c0(κ). These results are obtained as special cases of properties of “*-paired Banach spaces,” a notion generalizing the relation of a reflexive Banach space and its dual
Sobolev–Orlicz Imbeddings, Weak Compactness, and Spectrum

AbstractThe aim of this wok is to show how the weak compactness in the L1(X, m) space may be used to relate the existence of a Sobolev–Orlicz imbedding to the L2(X, m)-spectral properties of an operator H. In the first part we show that a Sobolev–Orlicz imbedding implies that the bottom of the L2-spectrum of H is an eigenvalue (i.e. the existence of the ground state) with finite multiplicity, provided m is finite. In the second part we prove that for a large class of operators, namely those for which Persson's characterization of the bottom of the essential spectrum holds true, a Sobolev–Orlicz imbedding always implies the discreteness of the L2-spectrum of H, provided m is finite. In the third part we show a certain converse of this last result in the sense that the discreteness of the L2-spectrum of H always implies the existence of an Orlicz space for which a Sobolev–Orlicz imbedding holds true for H. The case of logarithmic Sobolev inequalities is considered and provides the original motivations for this research
The Structure of Shift-Invariant Subspaces of L2(Rn)

AbstractUsing the range function approach to shift invariant spaces in L2(Rn) we give a simple characterization of frames and Riesz families generated by shifts of a countable set of generators in terms of their behavior on subspaces of l2(Zn). This in turn gives a simplified approach to the analysis of frames and Riesz families done by Gramians and dual Gramians. We prove a decomposition of a shift invariant space into the orthogonal sum of spaces each of which is generated by a quasi orthogonal generator. As an application of this fact we characterize shift preserving operators in terms of range operators and prove some facts about the dimension function
Lipschitz Algebras and Derivations II. Exterior Differentiation

AbstractBasic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Wiener space, etc. Although the constructions differ, in each of these cases one can define a module of measurable 1-forms and a first-order exterior derivative. We give a general construction which applies to any metric space equipped with a σ-finite measure and produces the desired result in all of the above cases. It agrees with Cheeger's construction when the latter is defined. It also applies to an important class of Dirichlet spaces, where, however, the known first-order differential calculus in general differs from ours (although the two are related)
A Characterization of (Locally) Uniformly Convex Spaces in Terms of Relative Openness of Quotient Maps on the Unit Ball

AbstractRelative openness of quotient maps on the closed unit ball U of a normed linear space X is studied quantitatively. Particularly, it follows from the results that the quotient maps on X associated with the closed linear subspaces of X are equally relatively open on U if and only if X is locally uniformly convex. Also, X is locally uniformly convex if and only if for any family of linear maps defined on X, equal relative openness on X implies equal relative openness on U. Similarly, uniformly convex spaces can be characterized in terms of equal uniform relative openness of quotient maps on U
Existence of Solutions of Stationary Compressible Navier–Stokes Equations with Large Force

AbstractIn this paper, we consider the Navier–Stokes equations for isentropic, compressible flows of a polytropic gas in a bounded domain. The equations to be considered are obtained by scaling to dimensionless form and then replacing the density ρ by ρ+ε2ρ, where ε is a Mach number. The existence of solutions has been known only for small forces or large potential forces near a rest state. The aim of this paper is to give a proof of the existence of stationary compressible Navier–Stokes equations with large force, when the Mach number ε is small
A Forelli–Rudin Construction and Asymptotics of Weighted Bergman Kernels

AbstractLet Ω be a pseudoconvex domain in CN with smooth boundary, −φ, −ψ two smooth defining functions for Ω={φ>0} such that −logψ, −logφ are plurisubharmonic, z∈Ω a point at which −logφ is strictly plurisubharmonic, and M⩾0 an integer. We show that as k→∞, the Bergman kernels with respect to the weights φkψM have the asymptotic expansionKφkψM(z, z)=kNπNφ(z)kψ(z)M∑j=0∞bj(z)k−j,b0=det−∂2logφ∂zj∂zk.For Ω strongly pseudoconvex with real-analytic boundary, φ, ψ real analytic and −logφ, −logψ strictly plurisubharmonic on Ω, we obtain also the analogous result for KφkψM(x, y) for (x, y) near the diagonal and discuss consequences for the asymptotics of the Berezin transform and for the Berezin quantization. The proofs rely on Fefferman's expansion for the Bergman kernel in a certain Forelli–Rudin type domain over Ω; as another application, they also yield a generalization of the cited Fefferman's expansion to a class of weighted Bergman kernels
Equivalence between Regularity Theorems and Heat Kernel Estimates for Higher Order Elliptic Operators and Systems under Divergence Form

AbstractWe study the heat kernel of higher order elliptic operators or systems under divergence form on Rn. Ellipticity is in the sense of Gårding inequality. We show that for homogeneous operators Gaussian upper bounds and Hölder regularity of the heat kernel is equivalent to local regularity of weak solutions. We also show stability of such bounds under L∞-perturbations of the coefficients or under perturbations with bounded coefficients lower order terms. Such a criterion allows us to obtain heat kernel bounds for operators or systems with uniformly continuous or vmo coefficients
On the Scattering Length Spectrum for Real Analytic Obstacles

AbstractIt follows trivially from old results of Majda and Lax–Phillips that connected obstacles K with real analytic boundary in Rn are uniquely determined by their scattering length spectrum. In this paper we prove a similar result in the general case (i.e. K may be disconnected) imposing some non-degeneracy conditions on K and assuming that its trapping set does not topologically divide S*(C), where C is a sphere containing K. It is shown that the conditions imposed on K are fulfilled, for instance, when K is a finite disjoint union of strictly convex bodies
Reflection Symmetries of Almost Periodic Functions

AbstractWe study global reflection symmetries of almost periodic functions. In the non-limit periodic case, we establish an upper bound on the Haar measure of the set of those elements in the hull which are almost symmetric about the origin. As an application of this result we prove that in the non-limit periodic case, the criterion of Jitomirskaya and Simon ensuring absence of eigenvalues for almost periodic Schrödinger operators is only applicable on a set of zero Haar measure. We complement this by giving examples of limit periodic functions where the Jitomirskaya–Simon criterion can be applied to every element of the hull
A Weakly Polyhomogeneous Calculus for Pseudodifferential Boundary Problems

AbstractIn a joint work with R. Seeley, a calculus of weakly parametric pseudodifferential operators on closed manifolds was introduced and used to obtain complete asymptotic expansions of traces of resolvents and heat operators associated with the Atiyah–Patodi–Singer problem. The present paper establishes a generalization to pseudodifferential boundary operators, defining weakly polyhomogeneous singular Green operators, Poisson operators, and trace operators associated with a manifold with boundary, as well as a suitable transmission condition for pseudodifferential operators. Full composition formulas are established for the calculus, which contains the resolvents of APS-type problems. The operators in the calculus have complete asymptotic trace expansions in the parameter (when of trace class), with polynomial and logarithmic terms
A Galois Correspondence for II1 Factors and Quantum Groupoids

AbstractWe establish a Galois correspondence for finite quantum groupoid actions on II1 factors and show that every finite index and finite depth subfactor is an intermediate subalgebra of a quantum groupoid crossed product. Moreover, any such subfactor is completely and canonically determined by a quantum groupoid and its coideal *-subalgebra. This allows us to express the bimodule category of a subfactor in terms of the representation category of a corresponding quantum groupoid and the principal graph as the Bratteli diagram of an inclusion of certain finite-dimensional C*-algebras related to it
Strongly Asymptotic Morphisms on Separable Metrisable Algebras

AbstractAsymptotic morphisms on C* algebras and their compositions were introduced by Connes and Higson. This paper considers definitions of asymptotic morphisms on separable metrisable algebras, and a compatibility condition is given which allows the composition of such morphisms. A class of algebras is defined, with the property that every bounded set has compact closure, where the compatibility conditions are automatically satisfied. Three examples are given in detail, the first involving a non-normable algebra due to Elliott, Natsume and Nest. The second is integration with respect to a certain quasi-commutative spectral measure on the algebra of paths on a C* algebra, and the third the equivalence between the suspensions of the mapping cone and the ideal for a short exact sequence of C* algebras
Continuous Functions on Totally Ordered Spaces That Are Compact in Their Order Topologies

AbstractThe totally ordered spaces that are compact and connected in their order topologies are characterized as images of lexicographic cubes. The Banach space of continuous functions on the lexicographic product of compact totally ordered spaces, whose spaces of continuous functions have locally uniformly convex norms, has an equivalent locally uniformly convex norm if; and only if, the product is countable. The Banach space of continuous functions on a compact totally ordered space always has an equivalent Kadec norm. Those compact totally ordered spaces, for which the Banach space of continuous functions has an equivalent locally uniformly convex norm, are characterized in terms of the bounded decreasing interval functions that can be defined on the intervals of the totally ordered spaces. Some examples are discussed in detail
Inclusions of Von Neumann Algebras and Quantum Groupoı&#x0308;ds II

AbstractIn a previous article, in collaboration with Jean-Michel Vallin, we constructed two quantum groupoı&#x0308;ds, dual to each other, from a depth 2 inclusion of von Neumann algebras M0⊂M1. In this paper we investigate this structure in greater detail. In the previous article, we constructed the analog of a co-product, while in this paper we define a co-inverse, by making the polar decomposition of the analog of the antipode, and left and right invariant Haar operator-valued weights. These two structures of quantum groupoı&#x0308;ds, dual to each other, can be placed on the relative commutants M′0∩M2 and M′1∩M3 in such a way that the canonical Jones' tower associated to the inclusion can be described as a tower of successive crossed-products by these two structures
Morera Theorems for Complex Manifolds

AbstractWe prove Morera theorems for the Radon transform integrating on geodesic spheres on complex analytic manifolds of arbitrary dimension. To avoid pathologies, we assume that the radius of each sphere of integration is less than the injectivity radius at its center. The proofs of the main results are local, and they involve the microlocal properties of associated Radon transforms and a theorem of Hörmander, Kawai, and Kashiwara on microlocal singularities. We consider Morera theorems for spheres of fixed radius and spheres of arbitrary radius