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  • Corrigendum to “Trace spaces of counterexamples to Naimark's problem” [J. Funct. Anal. 275 (10) (2018) 2794–2816]
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-20
    Andrea Vaccaro

    Because of a mistake in the proof of [10, Theorem A - part 1], the main statements of [10] ([10, Theorem 1 - part 1] and [10, Theorem 2]) are not proved in full generality. We provide an alternative proof to such statements.

    更新日期:2020-01-21
  • Topological centres of weighted convolution algebras
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-14
    Mahmoud Filali; Pekka Salmi

    Let G be a non-compact locally compact group with a continuous submultiplicative weight function ω such that ω(e)=1 and ω is diagonally bounded with bound K≥1. When G is σ-compact, we show that ⌊K⌋+1 many points in the spectrum of LUC(ω−1) are enough to determine the topological centre of LUC(ω−1)⁎ and that ⌊K⌋+2 many points in the spectrum of L∞(ω−1) are enough to determine the topological centre of L1(ω)⁎⁎ when G is in addition a SIN-group. We deduce that the topological centre of LUC(ω−1)⁎ is the weighted measure algebra M(ω) and that of C0(ω−1)⊥ is trivial for any locally compact group. The topological centre of L1(ω)⁎⁎ is L1(ω) and that of of L0∞(ω)⊥ is trivial for any non-compact locally compact SIN-group. The same techniques apply and lead to similar results when G is a weakly cancellative right cancellative discrete semigroup.

    更新日期:2020-01-15
  • Incomplete Yamabe flows and removable singularities
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-14
    Mario B. Schulz

    We study the Yamabe flow on a Riemannian manifold of dimension m≥3 minus a closed submanifold of dimension n and prove that there exists an instantaneously complete solution if and only if n>m−22. In the remaining cases 0≤n≤m−22 including the borderline case, we show that the removability of the n-dimensional singularity is necessarily preserved along the Yamabe flow. In particular, the flow must remain geodesically incomplete as long as it exists. This is contrasted with the two-dimensional case, where instantaneously complete solutions always exist.

    更新日期:2020-01-15
  • A Bernstein type theorem for minimal hypersurfaces via Gauss maps
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-14
    Qi Ding

    Let M be an n-dimensional smooth oriented complete embedded minimal hypersurface in Rn+1 with Euclidean volume growth. We show that if the image under the Gauss map of M avoids some neighborhood of a half-equator, then M must be an affine hyperplane.

    更新日期:2020-01-15
  • A characterization of modulation spaces by symplectic rotations
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-14
    Elena Cordero; Maurice de Gosson; Fabio Nicola

    This note contains a new characterization of modulation spaces Mmp(Rn), 1≤p≤∞, by symplectic rotations. Precisely, instead to measure the time-frequency content of a function by using translations and modulations of a fixed window as building blocks, we use translations and metaplectic operators corresponding to symplectic rotations. Technically, this amounts to replace, in the computation of the Mmp(Rn)-norm, the integral in the time-frequency plane with an integral on Rn×U(2n,R) with respect to a suitable measure, U(2n,R) being the group of symplectic rotations. More conceptually, we are considering a sort of polar coordinates in the time-frequency plane. To have invariance under symplectic rotations we choose a Gaussian as suitable window function. We also provide a similar (and easier) characterization with the group U(2n,R) being reduced to the n-dimensional torus Tn.

    更新日期:2020-01-15
  • On the embeddability of the family of countably branching trees into quasi-reflexive Banach spaces
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-14
    Y. Perreau

    In this note we extend to the quasi-reflexive setting the result of F. Baudier, N. Kalton and G. Lancien concerning the non-embeddability of the family of countably branching trees into reflexive Banach spaces whose Szlenk index and Szlenk index from the dual are both equal to the first infinite ordinal ω. In particular we show that the family of countably branching trees does neither embed into the James space Jp nor into its dual space Jp⁎ for p∈(1,∞).

    更新日期:2020-01-15
  • Tempered distributions and Schwartz functions on definable manifolds
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-14
    Ary Shaviv

    We define the spaces of Schwartz functions, tempered functions and tempered distributions on manifolds definable in polynomially bounded o-minimal structures. We show that all the classical properties that these spaces have in the Nash category, as first studied in Fokko du Cloux's work, also hold in this generalized setting. We also show that on manifolds definable in o-minimal structures that are not polynomially bounded, such a theory can not be constructed. We present some possible applications, mainly in representation theory.

    更新日期:2020-01-15
  • Difference equations and pseudo-differential operators on Zn
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-14
    Linda N.A. Botchway; P. Gaël Kibiti; Michael Ruzhansky

    In this paper we develop the calculus of pseudo-differential operators on the lattice Zn, which we can call pseudo-difference operators. An interesting feature of this calculus is that the global frequency space (Tn) is compact so the symbol classes are defined in terms of the behaviour with respect to the lattice variable. We establish formulae for composition, adjoint, transpose, and for parametrix for the elliptic operators. We also give conditions for the ℓ2, weighted ℓ2, and ℓp boundedness of operators and for their compactness on ℓp. We describe a link to the toroidal quantization on the torus Tn, and apply it to give conditions for the membership in Schatten classes on ℓ2(Zn). Furthermore, we discuss a version of Fourier integral operators on the lattice and give conditions for their ℓ2-boundedness. The results are applied to give estimates for solutions to difference equations on the lattice Zn. Moreover, we establish Gårding and sharp Gårding inequalities, with an application to the unique solvability of parabolic equations on the lattice Zn.

    更新日期:2020-01-15
  • Bianalytic free maps between spectrahedra and spectraballs
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-14
    J. William Helton; Igor Klep; Scott McCullough; Jurij Volčič

    Linear matrix inequalities (LMIs) are ubiquitous in real algebraic geometry, semidefinite programming, control theory and signal processing. LMIs with (dimension free) matrix unknowns are central to the theories of completely positive maps and operator algebras, operator systems and spaces, and serve as the paradigm for matrix convex sets. The matricial feasibility set of an LMI is called a free spectrahedron. In this article, the bianalytic maps between a very general class of ball-like free spectrahedra (examples of which include row or column contractions, and tuples of contractions) and arbitrary free spectrahedra are characterized and seen to have an elegant algebraic form. They are all highly structured rational maps. In the case that both the domain and codomain are ball-like, these bianalytic maps are explicitly determined and the article gives necessary and sufficient conditions for the existence of such a map with a specified value and derivative at a point. In particular, this result leads to a classification of automorphism groups of ball-like free spectrahedra. The proofs depend on a novel free Nullstellensatz, established only after new tools in free analysis are developed and applied to obtain fine detail, geometric in nature locally and algebraic in nature globally, about the boundary of ball-like free spectrahedra.

    更新日期:2020-01-15
  • Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-10
    Patricia Alonso Ruiz; Fabrice Baudoin; Li Chen; Luke G. Rogers; Nageswari Shanmugalingam; Alexander Teplyaev

    We introduce heat semigroup-based Besov classes in the general framework of Dirichlet spaces. General properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are obtained. As a highlight of the paper, we obtain a far reaching Lp-analogue, p≥1, of the Sobolev inequality that was proved for p=2 by N. Varopoulos under the assumption of ultracontractivity for the heat semigroup. The case p=1 is of special interest since it yields isoperimetric type inequalities.

    更新日期:2020-01-11
  • Magic identities for the conformal four-point integrals; the Minkowski metric case
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-13
    Matvei Libine

    The original “magic identities” are due to J. M. Drummond, J. Henn, V. A. Smirnov and E. Sokatchev; they assert that all n-loop box integrals for four scalar massless particles are equal to each other [3]. The authors give a proof of the magic identities for the Euclidean metric case only and claim that the result is also true in the Minkowski metric. However, the Minkowski case is much more subtle and requires specification of the relative positions of cycles of integration to make these identities correct. In this article we prove the magic identities in the Minkowski metric case and, in particular, specify the cycles of integration. Our proof of magic identities relies on previous results from [7], [8], where we give a mathematical interpretation of the n-loop box integrals in the context of representations of a Lie group U(2,2) and quaternionic analysis. The main result of [7], [8] is a (weaker) operator version of the “magic identities”. No prior knowledge of physics or Feynman diagrams is assumed from the reader. We provide a summary of all relevant results from quaternionic analysis to make the article self-contained.

    更新日期:2020-01-04
  • Uniqueness properties of solutions to the Benjamin-Ono equation and related models
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-13
    C.E. Kenig; G. Ponce; L. Vega

    We prove that if u1,u2 are real solutions of the Benjamin-Ono equation defined in (x,t)∈R×[0,T] which agree in an open set Ω⊂R×[0,T], then u1≡u2. We extend this uniqueness result to a general class of equations of Benjamin-Ono type in both the initial value problem and the initial periodic boundary value problem. This class of 1-dimensional non-local models includes the intermediate long wave equation. We relate our uniqueness results with those for a water wave problem. Finally, we present a slightly stronger version of our uniqueness results for the Benjamin-Ono equation.

    更新日期:2020-01-04
  • An extension problem related to the fractional Branson–Gover operators
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-13
    Jan Frahm; Bent Ørsted; Genkai Zhang

    The Branson–Gover operators are conformally invariant differential operators of even degree acting on differential forms. They can be interpolated by a holomorphic family of conformally invariant integral operators called fractional Branson–Gover operators. For Euclidean spaces we show that the fractional Branson–Gover operators can be obtained as Dirichlet-to-Neumann operators of certain conformally invariant boundary value problems, generalizing the work of Caffarelli–Silvestre for the fractional Laplacians to differential forms. The relevant boundary value problems are studied in detail and we find appropriate Sobolev type spaces in which there exist unique solutions and obtain the explicit integral kernels of the solution operators as well as some of their properties.

    更新日期:2020-01-04
  • Pointwise gradient estimates for a class of singular quasilinear equations with measure data
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-12
    Quoc-Hung Nguyen; Nguyen Cong Phuc

    Local and global pointwise gradient estimates are obtained for solutions to the quasilinear elliptic equation with measure data −div(A(x,∇u))=μ in a bounded and possibly nonsmooth domain Ω in Rn. Here div(A(x,∇u)) is modeled after the p-Laplacian. Our results extend earlier known results to the singular case in which 3n−22n−1

    更新日期:2020-01-04
  • Characterization of initial data in the homogeneous Besov space for solutions in the Serrin class of the Navier-Stokes equations
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-13
    Hideo Kozono; Akira Okada; Senjo Shimizu

    Consider the Cauchy problem of the Navier-Stokes equations in Rn with initial data a in the homogeneous Besov space B˙p,q−1+np(Rn) for n

    更新日期:2020-01-04
  • Incompressible inhomogeneous fluids in bounded domains of R3 with bounded density
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-12
    Reinhard Farwig; Chenyin Qian; Ping Zhang

    In this paper, we study the incompressible inhomogeneous Navier-Stokes equations in bounded domains of R3 involving bounded density functions ρ=1+a. Based on the corresponding theory of Besov spaces on domains, we first obtain the global existence of weak solutions (ρ,u) with initial data a0∈L∞(Ω), u0∈Bq,s−1+3/q(Ω) for 1

    更新日期:2020-01-04
  • Prime II1 factors arising from actions of product groups
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-10-24
    Daniel Drimbe

    We prove that any II1 factor arising from a free ergodic probability measure preserving action Γ↷X of a product Γ=Γ1×…×Γn of icc hyperbolic, free product or wreath product groups is prime, provided Γi↷X is ergodic, for any 1≤i≤n. We also completely classify all the tensor product decompositions of a II1 factor associated to a free ergodic probability measure preserving action of a product of icc, hyperbolic, property (T) groups. As a consequence, we derive a unique prime factorization result for such II1 factors. Finally, we obtain a unique prime factorization theorem for a large class of II1 factors which have property Gamma.

    更新日期:2020-01-04
  • Difference of weighted composition operators
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-13
    Boo Rim Choe; Koeun Choi; Hyungwoon Koo; Jongho Yang

    We obtain complete characterizations in terms of Carleson measures for bounded/compact differences of weighted composition operators acting on the standard weighted Bergman spaces over the unit disk. Unlike the known results, we allow the weight functions to be non-holomorphic and unbounded. As a consequence we obtain a compactness characterization for differences of unweighted composition operators acting on the Hardy spaces in terms of Carleson measures and, as a nontrivial application of this, we show that compact differences of composition operators with univalent symbols on the Hardy spaces are exactly the same as those on the weighted Bergman spaces. As another application, we show that an earlier characterization due to Acharyya and Wu for compact differences of weighted composition operators with bounded holomorphic weights does not extend to the case of non-holomorphic weights. We also include some explicit examples related to our results.

    更新日期:2020-01-04
  • Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-13
    Anton Savin; Elmar Schrohe

    We consider elliptic operators associated with discrete groups of quantized canonical transformations. In order to be able to apply results from algebraic index theory, we define the localized algebraic index of the complete symbol of an elliptic operator. With the help of a calculus of semiclassical quantized canonical transformations, a version of Egorov's theorem and a theorem on trace asymptotics for semiclassical Fourier integral operators we show that the localized analytic index and the localized algebraic index coincide. As a corollary, we express the Fredholm index in terms of the algebraic index for a wide class of groups, in particular, for finite extensions of Abelian groups.

    更新日期:2020-01-04
  • Inversion problem in measure and Fourier–Stieltjes algebras
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-13
    Przemysław Ohrysko; Mateusz Wasilewski

    In this paper we study the inversion problem in measure and Fourier–Stieltjes algebras from qualitative and quantitative point of view extending the results obtained by N. Nikolski in [10].

    更新日期:2020-01-04
  • Compact linear combination of composition operators on Bergman spaces
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-13
    Boo Rim Choe; Hyungwoon Koo; Maofa Wang

    Motivated by the question of Shapiro and Sundberg raised in 1990, study on linear combinations of composition operators has been a topic of growing interest. In this paper, we completely characterize the compactness of any finite linear combination of composition operators with general symbols on the weighted Bergman spaces in two classical terms: one is a function theoretic characterization of Julia-Carathéodory type and the other is a measure theoretic characterization of Carleson type. Our approach is completely different from what has been known so far.

    更新日期:2020-01-04
  • Bibasic sequences in Banach lattices
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-03
    M.A. Taylor; V.G. Troitsky

    Given a Schauder basic sequence (xk) in a Banach lattice, we say that (xk) is bibasic if the expansion of every vector in [xk] converges not only in norm, but also in order. We prove that, in this definition, order convergence may be replaced with uniform convergence, with order boundedness of the partial sums, or with norm boundedness of finite suprema of the partial sums. The results in this paper extend and unify those from the pioneering paper Order Schauder bases in Banach lattices by A. Gumenchuk, O. Karlova, and M. Popov. In particular, we are able to characterize bibasic sequences in terms of the bibasis inequality, a result they obtained under certain additional assumptions. After establishing the aforementioned characterizations of bibasic sequences, we embark on a deeper study of their properties. We show, for example, that they are independent of ambient space, stable under small perturbations, and preserved under sequentially uniformly continuous norm isomorphic embeddings. After this we consider several special kinds of bibasic sequences, including permutable sequences, i.e., sequences for which every permutation is bibasic, and absolute sequences, i.e., sequences where expansions remain convergent after we replace every term with its modulus. We provide several equivalent characterizations of absolute sequences, showing how they relate to bibases and to further modifications of the basis inequality. We further consider bibasic sequences with unique order expansions. We show that this property does generally depend on ambient space, but not for the inclusion of c0 into ℓ∞. We also show that small perturbations of bibases with unique order expansions have unique order expansions, but this is not true if “bibases” is replaced with “bibasic sequences”. In the final section, we consider uo-bibasic sequences, which are obtained by replacing order convergence with uo-convergence in the definition of a bibasic sequence. We show that such sequences are very common.

    更新日期:2020-01-04
  • Norm-square localization and the quantization of Hamiltonian loop group spaces
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-03
    Yiannis Loizides; Yanli Song

    In an earlier article we introduced a new definition for the ‘quantization’ of a Hamiltonian loop group space M, involving the equivariant L2-index of a Dirac-type operator D on a non-compact finite dimensional submanifold Y of M. In this article we study a deformation of this operator, similar to the work of Tian-Zhang and Ma-Zhang. We obtain a formula for the index with infinitely many non-trivial contributions, indexed by the components of the critical set of the norm-square of the moment map. This is the main part of a new proof of the [Q,R]=0 theorem for Hamiltonian loop group spaces.

    更新日期:2020-01-04
  • Weak and strong type estimates for the multilinear pseudo-differential operators
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-03
    Mingming Cao; Qingying Xue; Kôzô Yabuta

    In this paper, we investigate the boundedness of the multilinear pseudo-differential operator Tσ. First, we establish the local exponential decay estimates for Tσ. In terms of the corresponding commutators Tσ,Σb, we obtain the local subexponential decay estimates. Secondly, we derive the weighted mixed weak type inequality for Tσ, which parallels Sawyer's conjecture for Calderón-Zygmund operators and covers the endpoint weighted inequalities. Last but not least, we present the sharp weighted estimates for Tσ and Tσ,Σb. It is worth mentioning that our results are totally new even in the linear case.

    更新日期:2020-01-04
  • Operator-valued chordal Loewner chains and non-commutative probability
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-03
    David Jekel

    We adapt the theory of chordal Loewner chains to the operator-valued matricial upper-half plane over a C⁎-algebra A. We define an A-valued chordal Loewner chain as a subordination chain of analytic self-maps of the A-valued upper half-plane, such that each Ft is the reciprocal Cauchy transform of an A-valued law μt, such that the mean and variance of μt are continuous functions of t. We relate A-valued Loewner chains to processes with A-valued free or monotone independent independent increments just as was done in the scalar case by Bauer [1] and Schleißinger [2]. We show that the Loewner equation ∂tFt(z)=DFt(z)[Vt(z)], when interpreted in a certain distributional sense, defines a bijection between Lipschitz mean-zero Loewner chains Ft and vector fields Vt(z) of the form Vt(z)=−Gνt(z) where νt is a generalized A-valued law. Based on the Loewner equation, we derive a combinatorial expression for the moments of μt in terms of νt. We also construct non-commutative random variables on an operator-valued monotone Fock space which realize the laws μt. Finally, we prove a version of the monotone central limit theorem which describes the behavior of Ft as t→+∞ when νt has uniformly bounded support.

    更新日期:2020-01-04
  • Radiation condition bounds on manifolds with ends
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-03
    K. Ito; E. Skibsted

    We study spectral theory for the Schrödinger operator on manifolds possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends. Certain exterior domains for possibly unbounded obstacles are included. We prove Rellich's theorem, the limiting absorption principle, radiation condition bounds and the Sommerfeld uniqueness result, striving to extending and refining previously known spectral results on manifolds. The proofs are given by an extensive use of commutator arguments. These arguments have a classical spirit (essentially) not involving energy cutoffs or microlocal analysis and require, presumably, minimum regularity and decay properties of perturbations. This paper has interest of its own right, but it also serves as a basis for the stationary scattering theory developed fully in the sequel [20].

    更新日期:2020-01-04
  • On boundedness and compactness of Toeplitz operators in weighted H∞-spaces
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-02
    José Bonet; Wolfgang Lusky; Jari Taskinen

    We characterize the boundedness and compactness of Toeplitz operators Ta with radial symbols a in weighted H∞-spaces Hv∞ on the open unit disc of the complex plane. The weights v are also assumed radial and to satisfy the condition (B) introduced by the second named author. The main technique uses Taylor coefficient multipliers, and the results are first proved for them. We formulate a related sufficient condition for the boundedness and compactness of Toeplitz operators in reflexive weighted Bergman spaces on the disc. We also construct a bounded harmonic symbol f such that Tf is not bounded in Hv∞ for any v satisfying mild assumptions. As a corollary, the Bergman projection is never bounded with respect to the corresponding weighted sup-norms. However, we also show that, for normal weights v, all Toeplitz operators with a trigonometric polynomial as the symbol are bounded on Hv∞.

    更新日期:2020-01-04
  • Superlinear elliptic inequalities on manifolds
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-02
    Alexander Grigor'yan; Yuhua Sun; Igor Verbitsky

    Let M be a complete non-compact Riemannian manifold and let σ be a Radon measure on M. We study the problem of existence or non-existence of positive solutions to a semilinear elliptic inequaliy−Δu≥σuqinM, where q>1. We obtain necessary and sufficent criteria for existence of positive solutions in terms of Green function of Δ. In particular, explicit necessary and sufficient conditions are given when M has nonnegative Ricci curvature everywhere in M, or more generally when Green's function satisfies the 3G-inequality.

    更新日期:2020-01-04
  • Asymptotics of Cheeger constants and unitarisability of groups
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-02
    Maria Gerasimova; Dominik Gruber; Nicolas Monod; Andreas Thom

    Given a group Γ, we establish a connection between the unitarisability of its uniformly bounded representations and the asymptotic behaviour of the isoperimetric constants of Cayley graphs of Γ for increasingly large generating sets. The connection hinges on an analytic invariant Lit(Γ)∈[0,∞] which we call the Littlewood exponent. Finiteness, amenability, unitarisability and the existence of free subgroups are related respectively to the thresholds 0,1,2 and ∞ for Lit(Γ). Using graphical small cancellation theory, we prove that there exist groups Γ for which 1

    更新日期:2020-01-04
  • Nonlinear operations on a class of modulation spaces
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-02
    Tomoya Kato; Mitsuru Sugimoto; Naohito Tomita

    We discuss when the nonlinear operation f↦F(f) maps the modulation space Msp,q(Rn) (1≤p,q≤∞) to the same space again. It is known that Msp,q(Rn) is a multiplication algebra when s>n−n/q, hence it is true for this space if F is entire. We claim that it is still true for non-analytic F when q≥4/3.

    更新日期:2020-01-04
  • Spectral enclosures for a class of block operator matrices
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-02
    Juan Giribet; Matthias Langer; Francisco Martínez Pería; Friedrich Philipp; Carsten Trunk

    We prove new spectral enclosures for the non-real spectrum of a class of 2×2 block operator matrices with self-adjoint operators A and D on the diagonal and operators B and −B⁎ as off-diagonal entries. One of our main results resembles Gershgorin's circle theorem. The enclosures are applied to J-frame operators.

    更新日期:2020-01-04
  • A formula for the anisotropic total variation of SBV functions
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-02
    Fernando Farroni; Nicola Fusco; Serena Guarino Lo Bianco; Roberta Schiattarella

    The purpose of this paper is to present the relation between certain BMO–type seminorms and the total variation of SBV functions. Following some ideas of [2], we give a representation formula of the total variation of SBV functions which does not make use of the distributional derivatives. We consider an anisotropic variant of the BMO–type seminorm introduced in [4], by using, instead of cubes, covering families made by translations of a given open bounded set with Lipschitz boundary.

    更新日期:2020-01-04
  • Finite field restriction estimates for the paraboloid in high even dimensions
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-02
    Alex Iosevich; Doowon Koh; Mark Lewko

    We prove that the finite field Fourier extension operator for the paraboloid is bounded from L2→Lr for r≥2d+4d in even dimensions d≥8, which is the optimal L2 estimate. For d=6 we obtain the optimal range r>2d+4d=8/3, apart from the endpoint. For d=4 we improve the prior range of r>16/5=3.2 to r≥28/9=3.111…, compared to the conjectured range of r≥3. The key new ingredient is improved additive energy estimates for subsets of the paraboloid.

    更新日期:2020-01-04
  • Lifting for manifold-valued maps of bounded variation
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-02
    Giacomo Canevari; Giandomenico Orlandi

    Let N be a smooth, compact, connected Riemannian manifold without boundary. Let E→N be the Riemannian universal covering of N. For any bounded, smooth domain Ω⊆Rd and any u∈BV(Ω,N), we show that u has a lifting v∈BV(Ω,E). Our result proves a conjecture by Bethuel and Chiron.

    更新日期:2020-01-04
  • Distribution of scattering resonances for generic Schrödinger operators
    J. Funct. Anal. (IF 1.637) Pub Date : 2020-01-02
    Tien-Cuong Dinh; Viêt-Anh Nguyên

    Let −Δ+V be the Schrödinger operator acting on L2(Rd,C) with d≥3 odd. Here V is a bounded real- or complex-valued function vanishing outside the closed ball of center 0 and radius a. If V belongs to the class Ma of potentials introduced by Christiansen, we show that when r→∞, the resonances of −Δ+V, scaled down by the factor r, are asymptotically distributed, with respect to an explicit probability distribution on the lower unit half-disc of the complex plane. The rate of convergence is also considered for subclasses of potentials.

    更新日期:2020-01-04
  • Schauder estimates for drifted fractional operators in the supercritical case
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-12-19
    Paul-Éric Chaudru de Raynal; Stéphane Menozzi; Enrico Priola

    We consider a non-local operator Lα which is the sum of a fractional Laplacian △α/2, α∈(0,1), plus a first order term which is measurable in the time variable and locally β-Hölder continuous in the space variables. Importantly, the fractional Laplacian Δα/2 does not dominate the first order term. We show that global parabolic Schauder estimates hold even in this case under the natural condition α+β>1. Thus, the constant appearing in the Schauder estimates is in fact independent of the L∞-norm of the first order term. In our approach we do not use the so-called extension property and we can replace △α/2 with other operators of α-stable type which are somehow close, including the relativistic α-stable operator. Moreover, when α∈(1/2,1), we can prove Schauder estimates for more general α-stable type operators like the singular cylindrical one, i.e., when △α/2 is replaced by a sum of one dimensional fractional Laplacians ∑k=1d(∂xkxk2)α/2.

    更新日期:2020-01-04
  • On families of optimal Hardy-weights for linear second-order elliptic operators
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-12-13
    Yehuda Pinchover; Idan Versano

    We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator using a one-dimensional reduction. More precisely, we first characterize all optimal Hardy-weights with respect to one-dimensional subcritical Sturm-Liouville operators on (a,b), ∞≤a

    更新日期:2020-01-04
  • Faber-Krahn type inequalities and uniqueness of positive solutions on metric measure spaces
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-12-13
    Anup Biswas; Janna Lierl

    We consider a general class of metric measure spaces equipped with a strongly local regular Dirichlet form and provide a lower bound on the hitting time probabilities of the associated Hunt process. Using these estimates we establish (i) a generalization of the classical Lieb's inequality, and (ii) uniqueness of nonnegative super-solutions to semi-linear elliptic equations on metric measure spaces. Finally, using heat-kernel estimates we generalize the local Faber-Krahn inequality recently obtained in [28] to local and non-local Dirichlet spaces.

    更新日期:2020-01-04
  • Decomposing algebraic m-isometric tuples
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-12-13
    Trieu Le

    We show that any m-isometric tuple of commuting algebraic operators on a Hilbert space can be decomposed as a sum of a spherical isometry and a commuting nilpotent tuple. Our approach applies as well to tuples of algebraic operators that are hereditary roots of polynomials in several variables.

    更新日期:2020-01-04
  • Automorphic equivalence within gapped phases in the bulk
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-12-13
    Alvin Moon; Yoshiko Ogata

    We develop a new adiabatic theorem for unique gapped ground states which does not require the gap for local Hamiltonians. We instead require a gap in the bulk and a smoothness of expectation values of sub-exponentially localized observables in the unique gapped ground state φs(A). This requirement is weaker than the requirement of the gap of the local Hamiltonians, since a uniform spectral gap for finite dimensional ground states implies a gap in the bulk for unique gapped ground states, as well as the smoothness.

    更新日期:2020-01-04
  • An operator-valued T1 theory for symmetric CZOs
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-12-13
    Guixiang Hong; Honghai Liu; Tao Mei

    We provide a natural BMO-criterion for the L2-boundedness of Calderón-Zygmund operators with operator-valued kernels satisfying a symmetric property. Our arguments involve both classical and quantum probability theory. In the appendix, we give a proof of the L2-boundedness of the commutators [Rj,b] whenever b belongs to the Bourgain's vector-valued BMO space, where Rj is the j-th Riesz transform. A common ingredient is the operator-valued Haar multiplier studied by Blasco and Pott.

    更新日期:2020-01-04
  • Propagation in a Fisher-KPP equation with non-local advection
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-12-13
    François Hamel; Christopher Henderson

    We investigate the influence of a general non-local advection term of the form K⁎u to propagation in the one-dimensional Fisher-KPP equation. This model is a generalization of the Keller-Segel-Fisher system. When K∈L1(R), we obtain explicit upper and lower bounds on the propagation speed which are asymptotically sharp and more precise than previous works. When K∈Lp(R) with p>1 and is non-increasing in (−∞,0) and in (0,+∞), we show that the position of the “front” is of order O(tp) if p<∞ and O(eλt) for some λ>0 if p=∞ and K(+∞)>0. We use a wide range of techniques in our proofs.

    更新日期:2020-01-04
  • The Segal-Bargmann transform on classical matrix Lie groups
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-12-13
    Alice Z. Chan

    We study the complex-time Segal-Bargmann transform Bs,τKN on a compact type Lie group KN, where KN is one of the following classical matrix Lie groups: the special orthogonal group SO(N,R), the special unitary group SU(N), or the compact symplectic group Sp(N). Our work complements and extends the results of Driver, Hall, and Kemp on the Segal-Bargman transform for the unitary group U(N). We provide an effective method of computing the action of the Segal-Bargmann transform on trace polynomials, which comprise a subspace of smooth functions on KN extending the polynomial functional calculus. Using these results, we show that as N→∞, the finite-dimensional transform Bs,τKN has a meaningful limit Gs,τ which can be identified as an operator on the space of complex Laurent polynomials.

    更新日期:2020-01-04
  • On the convergence of stationary solutions in the Smoluchowski-Kramers approximation of infinite dimensional systems
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-12-13
    Sandra Cerrai; Nathan Glatt-Holtz

    We prove the convergence, in the small mass limit, of statistically invariant states for a class of semi-linear damped wave equations, perturbed by an additive Gaussian noise, both with Lipschitz-continuous and with polynomial non-linearities. In particular, we prove that the first marginals of any sequence of invariant measures for the stochastic wave equation converge in a suitable Wasserstein metric to the unique invariant measure of the limiting stochastic semi-linear parabolic equation obtained in the Smoluchowski-Kramers approximation. The Wasserstein metric is associated to a suitable distance on the space of square integrable functions, that is chosen in such a way that the dynamics of the limiting stochastic parabolic equation is contractive with respect to such a Wasserstein metric. This implies that the limiting result is a consequence of the validity of a generalized Smoluchowski-Kramers limit at fixed times. The proof of such a generalized limit requires new delicate bounds for the solutions of the stochastic wave equation, that must be uniform with respect to the size of the mass.

    更新日期:2020-01-04
  • Maximal function characterizations for Hardy spaces on spaces of homogeneous type with finite measure and applications
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-12-13
    The Anh Bui; Xuan Thinh Duong; Fu Ken Ly

    We prove nontangential and radial maximal function characterizations for Hardy spaces associated to a non-negative self-adjoint operator satisfying Gaussian estimates on a space of homogeneous type with finite measure. This not only addresses an open point in the literature, but also gives a complete answer to the question posed by Coifman and Weiss in the case of finite measure. We then apply our results to give maximal function characterizations for Hardy spaces associated to second–order elliptic operators with Neumann and Dirichlet boundary conditions, Schrödinger operators with Dirichlet boundary conditions, and Fourier–Bessel operators.

    更新日期:2020-01-04
  • Algebras of noncommutative functions on subvarieties of the noncommutative ball: The bounded and completely bounded isomorphism problem
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-12-13
    Guy Salomon; Orr M. Shalit; Eli Shamovich

    Given a noncommutative (nc) variety V in the nc unit ball Bd, we consider the algebra H∞(V) of bounded nc holomorphic functions on V. We investigate the problem of when two algebras H∞(V) and H∞(W) are isomorphic. We prove that these algebras are weak-⁎ continuously isomorphic if and only if there is an nc biholomorphism G:W˜→V˜ between the similarity envelopes that is bi-Lipschitz with respect to the free pseudo-hyperbolic metric. Moreover, such an isomorphism always has the form f↦f∘G, where G is an nc biholomorphism. These results also shed some new light on automorphisms of the noncommutative analytic Toeplitz algebras H∞(Bd) studied by Davidson–Pitts and by Popescu. In particular, we find that Aut(H∞(Bd)) is a proper subgroup of Aut(B˜d). When d<∞ and the varieties are homogeneous, we remove the weak-⁎ continuity assumption, showing that two such algebras are boundedly isomorphic if and only if there is a bi-Lipschitz nc biholomorphism between the similarity envelopes of the nc varieties. We provide two proofs. In the noncommutative setting, our main tool is the noncommutative spectral radius, about which we prove several new results. In the free commutative case, we use a new free commutative Nullstellensatz that allows us to bootstrap techniques from the fully commutative case.

    更新日期:2020-01-04
  • Some rigidity results for II1 factors arising from wreath products of property (T) groups
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-27
    Ionut Chifan; Bogdan Teodor Udrea

    We show that any infinite collection (Γn)n∈N of icc, hyperbolic, property (T) groups satisfies the following von Neumann algebraic infinite product rigidity phenomenon. If Λ is an arbitrary group such that L(⊕n∈NΓn)≅L(Λ) then there exists an infinite direct sum decomposition Λ=(⊕n∈NΛn)⊕A with A icc amenable or trivial such that, for all n∈N, up to amplifications, we have L(Γn)≅L(Λn) and L(⊕k≥nΓk)≅L((⊕k≥nΛk)⊕A). The result is sharp and complements the previous finite product rigidity property found in [16]. Using this we provide an uncountable family of restricted wreath products Γ≅Σ≀Δ of icc, property (T) groups Σ, Δ whose wreath product structure is recognizable, up to a normal amenable subgroup, from their von Neumann algebras L(Γ). Along the way we highlight several applications of these results to the study of rigidity in the C⁎-algebra setting.

    更新日期:2020-01-04
  • Matrix elements of irreducible representations of SU(n + 1)×SU(n + 1) and multivariable matrix-valued orthogonal polynomials
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-27
    Erik Koelink; Maarten van Pruijssen; Pablo Román

    In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup, so the spherical functions are matrix-valued. Under these assumptions these functions can be described in terms of matrix-valued orthogonal polynomials in several variables, where the number of variables is the rank of the compact symmetric pair. Moreover, these polynomials are uniquely determined as simultaneous eigenfunctions of a commutative algebra of differential operators. In Part 2 we verify that the group case SU(n+1) meets all the conditions that we impose in Part 1. For any k∈N0 we obtain families of orthogonal polynomials in n variables with values in the N×N-matrices, where N=(n+kk). The case k=0 leads to the classical Heckman-Opdam polynomials of type An with geometric parameter. For k=1 we obtain the most complete results. In this case we give an explicit expression of the matrix weight, which we show to be irreducible whenever n≥2. We also give explicit expressions of the spherical functions that determine the matrix weight for k=1. These expressions are used to calculate the spherical functions that determine the matrix weight for general k up to invertible upper-triangular matrices. This generalizes and gives a new proof of a formula originally obtained by Koornwinder for the case n=1. The commuting family of differential operators that have the matrix-valued polynomials as simultaneous eigenfunctions contains an element of order one. We give explicit formulas for differential operators of order one and two for (n,k) equal to (2,1) and (3,1).

    更新日期:2020-01-04
  • Existence of diametrically complete sets with empty interior in reflexive and separable Banach spaces
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-27
    Monika Budzyńska; Tadeusz Kuczumow; Simeon Reich; Mariola Walczyk

    In this paper we prove that every infinite-dimensional and separable Banach space (X,‖⋅‖X) admits an equivalent norm ‖⋅‖X,1 such that (X,‖⋅‖X,1) has both the Kadec-Klee and the Opial properties. This result also has a quantitative aspect and when combined with the properties of Schauder bases and the Day norm it constitutes a basic tool in the proof of our main theorem: each infinite-dimensional, reflexive and separable Banach space (X,‖⋅‖X) has an equivalent norm ‖⋅‖0 such that (X,‖⋅‖0) is LUR and contains a diametrically complete set with empty interior.

    更新日期:2020-01-04
  • Decay estimates for a dissipative-dispersive linear semigroup and application to the viscous Boussinesq equation
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-27
    Guowei Liu; Weike Wang

    At the core of this article is the new estimate for a class of dissipative-dispersive linear semigroup eΔt±ip(|∇|)t arising in the study of the viscous Boussinesq equation. We combine the decay estimate with introducing a set of time-weighted Sobolev spaces, where the time-weights and the regularity of the Sobolev spaces are determined by our decay estimate, to show the global existence and asymptotic behavior of solutions to the viscous Boussinesq equation in Rn.

    更新日期:2020-01-04
  • Higher order Sobolev trace inequalities on balls revisited
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-27
    Quốc Anh Ngô; Van Hoang Nguyen; Quoc Hung Phan

    Inspired by a recent sharp Sobolev trace inequality of order four on the balls Bn+1 found by Ache and Chang (2017) [2], we propose a different approach to reprove Ache–Chang's trace inequality. To further illustrate this approach, we reprove the classical Sobolev trace inequality of order two on Bn+1 and provide sharp Sobolev trace inequalities of orders six and eight on Bn+1. To obtain all these inequalities up to order eight, and possibly more, we first establish higher order sharp Sobolev trace inequalities on R+n+1, then directly transferring them to the ball via a conformal change. As the limiting case of the Sobolev trace inequalities, Lebedev–Milin type inequalities of order up to eight are also considered.

    更新日期:2020-01-04
  • Regular propagators of bilinear quantum systems
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-27
    Nabile Boussaïd; Marco Caponigro; Thomas Chambrion

    The present analysis deals with the regularity of solutions of bilinear control systems of the type x′=(A+u(t)B)x where the state x belongs to some complex infinite dimensional Hilbert space, the (possibly unbounded) linear operators A and B are skew-adjoint and the control u is a real valued function. Such systems arise, for instance, in quantum control with the bilinear Schrödinger equation. For the sake of the regularity analysis, we consider a more general framework where A and B are generators of contraction semigroups. Under some hypotheses on the commutator of the operators A and B, it is possible to extend the definition of solution for controls in the set of Radon measures to obtain precise a priori energy estimates on the solutions, leading to a natural extension of the celebrated noncontrollability result of Ball, Marsden, and Slemrod in 1982.

    更新日期:2020-01-04
  • The Steklov and Laplacian spectra of Riemannian manifolds with boundary
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-27
    Bruno Colbois; Alexandre Girouard; Asma Hassannezhad

    Given two compact Riemannian manifolds M1 and M2 such that their respective boundaries Σ1 and Σ2 admit neighbourhoods Ω1 and Ω2 which are isometric, we prove the existence of a constant C such that |σk(M1)−σk(M2)|≤C for each k∈N. The constant C depends only on the geometry of Ω1≅Ω2. This follows from a quantitative relationship between the Steklov eigenvalues σk of a compact Riemannian manifold M and the eigenvalues λk of the Laplacian on its boundary. Our main result states that the difference |σk−λk| is bounded above by a constant which depends on the geometry of M only in a neighbourhood of its boundary. The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant C is given explicitly in terms of bounds on the geometry of Ω1≅Ω2.

    更新日期:2020-01-04
  • Tensor algebras of product systems and their C⁎-envelopes
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-27
    Adam Dor-On; Elias Katsoulis

    Let (G,P) be an abelian, lattice ordered group and let X be a compactly aligned product system over P with coefficients in A. We show that the C*-envelope of the Nica tensor algebra NTX+ coincides with both Sehnem's covariance algebra A×XP and the co-universal C⁎-algebra NOXr for injective, gauge-compatible, Nica-covariant representations of Carlsen, Larsen, Sims and Vittadello. We give several applications of this result on both the selfadjoint and non-selfadjoint operator algebra theory. First we guarantee the existence of NOXr, thus settling a problem of Carlsen, Larsen, Sims and Vittadello which was open even for abelian, lattice ordered groups. As a second application, we resolve a problem posed by Skalski and Zacharias on dilating isometric representations of product systems to unitary representations. As a third application we characterize the C⁎-envelope of the tensor algebra of a finitely aligned higher-rank graph which also holds for topological higher-rank graphs. As a final application we prove reduced Hao-Ng isomorphisms for generalized gauge actions of discrete groups on C⁎-algebras of product systems. This generalizes recent results that were obtained by various authors in the case where (G,P)=(Z,N).

    更新日期:2020-01-04
  • Perturbations of Gibbs semigroups and the non-selfadjoint harmonic oscillator
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-27
    Lyonell Boulton

    Let T be the generator of a C0-semigroup e−Tt which is of trace class for all t>0 (a Gibbs semigroup). Let A be another closed operator, T-bounded with T-bound equal to zero. In general T+A might not be the generator of a Gibbs semigroup. In the first half of this paper we give sufficient conditions on A so that T+A is the generator of a Gibbs semigroup. We determine these conditions in terms of the convergence of the Dyson-Phillips expansion in suitable Schatten-von Neumann norms. In the second half of the paper we consider T=Hϑ=−e−iϑ∂x2+eiϑx2, the non-selfadjoint harmonic oscillator, on L2(R) and A=V, a locally integrable potential growing like |x|α at infinity for 0≤α<2. We establish that the Dyson-Phillips expansion converges in r Schatten-von Neumann norm in this case for r large enough and show that Hϑ+V is the generator of a Gibbs semigroup e−(Hϑ+V)τ for |arg⁡τ|≤π2−|ϑ|≠π2. From this we determine high energy asymptotics for the eigenvalues and the resolvent norm of Hϑ+V.

    更新日期:2020-01-04
  • Meromorphy of local zeta functions in smooth model cases
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-26
    Joe Kamimoto; Toshihiro Nose

    It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. But, in the case of general (C∞) smooth functions, the meromorphic extension problem is not obvious. Indeed, it has been recently shown that there exist specific smooth functions whose local zeta functions have singularities different from poles. In order to understand the situation of the meromorphic extension in the smooth case, we investigate a simple but essentially important case, in which the respective function is expressed as u(x,y)xayb+ flat function, where u(0,0)≠0 and a,b are nonnegative integers. After classifying flat functions into four types, we precisely investigate the meromorphic extension of local zeta functions in each case. Our results show new interesting phenomena in one of these cases. Actually, when a−1/a and their poles on the half-plane are contained in the set {−k/b:k∈Nwithk

    更新日期:2020-01-04
  • A Wiener test à la Landis for evolutive Hörmander operators
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-26
    Giulio Tralli; Francesco Uguzzoni

    In this paper we prove a Wiener-type characterization of boundary regularity, in the spirit of a classical result by Landis, for a class of evolutive Hörmander operators. We actually show the validity of our criterion for a larger class of degenerate-parabolic operators with a fundamental solution satisfying suitable two-sided Gaussian bounds. Our condition is expressed in terms of a series of balayages or, (as it turns out to be) equivalently, Riesz-potentials.

    更新日期:2020-01-04
  • Geometry of C⁎-algebras, and the bidual of their projective tensor product
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-25
    Matthias Neufang

    Given C⁎-algebras A and B, consider the Banach algebra A⊗γB, where ⊗γ denotes the projective Banach space tensor product. If A and B are commutative, this is the Varopoulos algebra VA,B; we write VA for VA,A. It has been an open problem for almost 40 years to determine precisely when A⊗γB is Arens regular; see, e.g., [33], [48], [49]. Even the commutative situation, in particular the case A=B=ℓ∞, has remained unsolved. We solve this classical question for arbitrary C⁎-algebras. Indeed, we show that A⊗γB is Arens regular if and only if A or B has the Phillips property; note that A has the latter property if and only if it is scattered and has the Dunford–Pettis Property. A further equivalent condition is that A⁎ has the Schur property, or, again equivalently, the enveloping von Neumann algebra A⁎⁎ is finite atomic, i.e., a direct sum of matrix algebras. Hence, Arens regularity of A⊗γB is entirely encoded in the geometry of the C⁎-algebras. In case A and B are von Neumann algebra, we conclude that A⊗γB is Arens regular (if and) only if A or B is finite-dimensional. We also show that this characterization does not generalize to the class of non-selfadjoint dual (even commutative) operator algebras. Specializing to commutative C⁎-algebras A and B, we obtain that VA,B is Arens regular if and only if A or B is scattered. We further describe the centre Z(VA⁎⁎), showing that it is Banach algebra isomorphic to A⁎⁎⊗ehA⁎⁎, where ⊗eh denotes the extended Haagerup tensor product. We deduce that VA is strongly Arens irregular (if and) only if A is finite-dimensional. Hence, VA is neither Arens regular nor strongly Arens irregular, if and only if A is non-scattered; as mentioned above, this is new even for the case A=ℓ∞.

    更新日期:2020-01-04
  • Amenability of Beurling algebras, corrigendum to a result in “Generalised notions of amenability, II” [J. Funct. Anal. 254 (7) (2008) 1776–1810]
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-20
    Fereidoun Ghahramani; Richard J. Loy; Yong Zhang

    We fix a gap in the proof of a result in our earlier paper “Generalised notions of amenability, II” (F. Ghahramani et al. (2008) [2]), and so provide a new proof to a characterization of amenability for Beurling algebras. The result answers a question raised by M.C. White (1991) [6].

    更新日期:2020-01-04
  • Gradient estimates for heat kernels and harmonic functions
    J. Funct. Anal. (IF 1.637) Pub Date : 2019-11-13
    Thierry Coulhon; Renjin Jiang; Pekka Koskela; Adam Sikora

    Let (X,d,μ) be a doubling metric measure space endowed with a Dirichlet form E deriving from a “carré du champ”. Assume that (X,d,μ,E) supports a scale-invariant L2-Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for p∈(2,∞]: (i) (Gp): Lp-estimate for the gradient of the associated heat semigroup; (ii) (RHp): Lp-reverse Hölder inequality for the gradients of harmonic functions; (iii) (Rp): Lp-boundedness of the Riesz transform (p<∞); (iv) (GBE): a generalised Bakry-Émery condition. We show that, for p∈(2,∞), (i), (ii) (iii) are equivalent, while for p=∞, (i), (ii), (iv) are equivalent. Moreover, some of these equivalences still hold under weaker conditions than the L2-Poincaré inequality. Our result gives a characterisation of Li-Yau's gradient estimate of heat kernels for p=∞, while for p∈(2,∞) it is a substantial improvement as well as a generalisation of earlier results by Auscher-Coulhon-Duong-Hofmann [7] and Auscher-Coulhon [6]. Applications to isoperimetric inequalities and Sobolev inequalities are given. Our results apply to Riemannian and sub-Riemannian manifolds as well as to non-smooth spaces, and to degenerate elliptic/parabolic equations in these settings.

    更新日期:2020-01-04
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