Abstract: Publication date: 15 May 2019Source: Journal of Functional Analysis, Volume 276, Issue 10Author(s): Assis Azevedo, Davide Azevedo, Mário Bessa, Maria Joana Torres In this paper we prove a weak version of Lusin's theorem for the space of Sobolev-(1,p) volume preserving homeomorphisms on closed and connected n-dimensional manifolds, n≥3, for pn this result is not true. More precisely, we obtain the density of Sobolev-(1,p) homeomorphisms in the space of volume preserving automorphisms, for the weak topology. Furthermore, the regularization of an automorphism in a uniform ball centered at the identity can be done in a Sobolev-(1,p) ball with the same radius centered at the identity.
Abstract: Publication date: 15 May 2019Source: Journal of Functional Analysis, Volume 276, Issue 10Author(s): Richard J. Smith, Stanimir Troyanski Relatively recently it was proved that if Γ is an arbitrary set, then any equivalent norm on c0(Γ) can be approximated uniformly on bounded sets by polyhedral norms and C∞ smooth norms, with arbitrary precision. We extend this result to more classes of spaces having uncountable symmetric bases, such as preduals of the ‘discrete’ Lorentz spaces d(w,1,Γ), and certain symmetric Nakano spaces and Orlicz spaces. We also show that, given an arbitrary ordinal number α, there exists a scattered compact space K having Cantor–Bendixson height at least α, such that every equivalent norm on C(K) can be approximated as above.
Abstract: Publication date: 15 May 2019Source: Journal of Functional Analysis, Volume 276, Issue 10Author(s): Clément Coine, Christian Le Merdy, Anna Skripka, Fedor Sukochev Let A be a selfadjoint operator in a separable Hilbert space, K a selfadjoint Hilbert–Schmidt operator, and f∈Cn(R). We establish that φ(t)=f(A+tK)−f(A) is n-times continuously differentiable on R in the Hilbert–Schmidt norm, provided either A is bounded or the derivatives f(i), i=1,…,n, are bounded. This substantially extends the results of [3] on higher order differentiability of φ in the Hilbert–Schmidt norm for f in a certain Wiener class. As an application of the second order S2-differentiability, we extend the Koplienko trace formula from the Besov class B∞12(R) [20] to functions f for which the divided difference f[2] admits a certain Hilbert space factorization.
Abstract: Publication date: 15 May 2019Source: Journal of Functional Analysis, Volume 276, Issue 10Author(s): Evgeni Dimitrov, Alisa Knizel We study a general class of log-gas ensembles on (shifted) quadratic lattices. We prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are Gaussian with a universal covariance. We apply our general results to analyze the asymptotic behavior of a q-boxed plane partition model introduced by Borodin, Gorin and Rains. In particular, we show that the global fluctuations of the height function on a fixed slice are described by a one-dimensional section of a pullback of the two-dimensional Gaussian free field.Our approach is based on a q-analogue of the Schwinger–Dyson (or loop) equations, which originate in the work of Nekrasov and his collaborators, and extends the methods developed by Borodin, Gorin and Guionnet to quadratic lattices.
Abstract: Publication date: 15 May 2019Source: Journal of Functional Analysis, Volume 276, Issue 10Author(s): Tommaso Bruno, Marco M. Peloso, Anita Tabacco, Maria Vallarino Let G be a noncompact connected Lie group, denote with ρ a right Haar measure and choose a family of linearly independent left-invariant vector fields X on G satisfying Hörmander's condition. Let χ be a positive character of G and consider the measure μχ whose density with respect to ρ is χ. In this paper, we introduce Sobolev spaces Lαp(μχ) adapted to X and μχ (1
Abstract: Publication date: 15 May 2019Source: Journal of Functional Analysis, Volume 276, Issue 10Author(s): Wojciech S. Ożański The surface growth model, ut+uxxxx+∂xxux2=0, is a one-dimensional fourth order equation, which shares a number of striking similarities with the three-dimensional incompressible Navier–Stokes equations, including the results regarding existence and uniqueness of solutions and the partial regularity theory. Here we show that a weak solution of this equation is smooth on a space-time cylinder Q if the Serrin condition ux∈Lq′Lq(Q) is satisfied, where q,q′∈[1,∞] are such that either 1/q+4/q′
Abstract: Publication date: 15 May 2019Source: Journal of Functional Analysis, Volume 276, Issue 10Author(s): Dongsheng Li, Shanshan Ma In this paper, we establish the existence of viscosity solutions of Hessian equations with singular right-hand sides and obtain the asymptotic boundary behavior of solutions. The asymptotic results generalize those for Poisson equations and Monge-Ampère equations, and are more precise than obtained from Hopf lemma.
Abstract: Publication date: 1 May 2019Source: Journal of Functional Analysis, Volume 276, Issue 9Author(s): Changliang Zhou, Chunqin Zhou We investigate a sharp Moser–Trudinger inequality which involves the anisotropic Dirichlet norm (∫ΩFN(∇u)dx)1N on W01,N(Ω) for N≥2. Here F is convex and homogeneous of degree 1, and its polar Fo represents a Finsler metric on RN. Under this anisotropic Dirichlet norm, we establish the Lions type concentration-compactness alternative. Then by using a blow-up procedure, we obtain the existence of extremal functions for this sharp geometric inequality.
Abstract: Publication date: 1 May 2019Source: Journal of Functional Analysis, Volume 276, Issue 9Author(s): Chunhe Li, Pengzi Miao, Zhizhang Wang Motivated by the quasi-local mass problem in general relativity, we study the rigidity of isometric immersions with the same mean curvature into a warped product space. As a corollary of our main result, two star-shaped hypersurfaces in a spatial Schwarzschild or AdS-Schwarzschild manifold with nonzero mass differ only by a rotation if they are isometric and have the same mean curvature. We also prove similar results if the mean curvature condition is replaced by an σ2-curvature condition.
Abstract: Publication date: 1 May 2019Source: Journal of Functional Analysis, Volume 276, Issue 9Author(s): Zihua Guo, Jinlu Li, Zhaoyang Yin We prove the inviscid limit of the incompressible Navier–Stokes equations in the same topology of Besov spaces as the initial data. The proof is based on proving the continuous dependence of the Navier–Stokes equations uniformly with respect to the viscosity. To show the latter, we rely on some Bona–Smith type argument in the Lp setting. Our obtained result implies a new result that the Cauchy problem of the Euler equations is locally well-posed in the borderline Besov space B∞,11(Rd), d≥2, in the sense of Hadmard, which is an open problem left in recent works by Bourgain and Li in [3], [4] and by Misiołek and Yoneda in [12], [13], [14].
Abstract: Publication date: 1 May 2019Source: Journal of Functional Analysis, Volume 276, Issue 9Author(s): Guy David, Joseph Feneuil, Svitlana Mayboroda In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension n−1 in Rn, and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Γ of dimension d in Rn, d
Abstract: Publication date: 1 May 2019Source: Journal of Functional Analysis, Volume 276, Issue 9Author(s): P. Ivanisvili, A. Volberg We show how Beckner's montonicity result on Hamming cube easily implies the monotonicity of a flow introduced by Janson in Hausdorff–Young inequality
Abstract: Publication date: 1 May 2019Source: Journal of Functional Analysis, Volume 276, Issue 9Author(s): Goro Akagi, Giulio Schimperna, Antonio Segatti This paper deals with the Cauchy–Dirichlet problem for the fractional Cahn–Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called Łojasiewicz–Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but C1) nonlinearities.
Abstract: Publication date: 1 May 2019Source: Journal of Functional Analysis, Volume 276, Issue 9Author(s): Wencai Liu In this paper, we consider the perturbed Stark operatorHu=H0u+qu=−u″−xu+qu, where q is the power-decaying perturbation. The criteria for q such that H=H0+q has at most one eigenvalue (finitely many, infinitely many eigenvalues) are obtained. All the results are quantitative and are generalized to the perturbed Stark type operator.
Abstract: Publication date: 1 May 2019Source: Journal of Functional Analysis, Volume 276, Issue 9Author(s): David Krejčiřík, Petr Siegl For one-dimensional Schrödinger operators with complex-valued potentials, we construct pseudomodes corresponding to large pseudoeigenvalues. We develop a first systematic non-semi-classical approach, which results in a substantial progress in achieving optimal conditions and conclusions as well as in covering a wide class of previously inaccessible potentials, including discontinuous ones. Applications of the present results to higher-dimensional Schrödinger operators are also discussed.
Abstract: Publication date: Available online 21 March 2019Source: Journal of Functional AnalysisAuthor(s): Shane Cooper, Marcus Waurick In this article we present a novel method for studying the asymptotic behaviour, with order-sharp error estimates, of the resolvents of parameter-dependent operator families. The method is applied to the study of differential equations with rapidly oscillating coefficients in the context of second-order PDE systems and the Maxwell system. This produces a non-standard homogenisation result that is characterised by ‘fibre-wise’ homogenisation of the related Floquet-Bloch PDEs. These fibre-homogenised resolvents are shown to be asymptotically equivalent to a whole class of operator families, including those obtained by standard homogenisation methods.
Abstract: Publication date: Available online 21 March 2019Source: Journal of Functional AnalysisAuthor(s): Hui Li, Dilian Yang Let G be a countable discrete amenable group, and Λ be a strongly connected finite k-graph. If (G,Λ) is a pseudo free and locally faithful self-similar action which satisfies the finite-state condition, then the structure of the KMS simplex of the C*-algebra OG,Λ associated to (G,Λ) is described: it is either empty or affinely isomorphic to the tracial state space of the C*-algebra of the periodicity group PerG,Λ of (G,Λ), depending on whether the Perron-Frobenius eigenvector of Λ preserves the G-action. As applications of our main results, we also exhibit several classes of important examples.
Abstract: Publication date: Available online 21 March 2019Source: Journal of Functional AnalysisAuthor(s): Sébastien Gouëzel, Vladimir Shchur There is a gap in the proof of the main theorem in the article [5] on optimal bounds for the Morse lemma in Gromov-hyperbolic spaces. We correct this gap, showing that the main theorem of [5] is true. We also describe a computer certification of this result.
Abstract: Publication date: Available online 20 March 2019Source: Journal of Functional AnalysisAuthor(s): Jacek Dziubański, Agnieszka Hejna For a normalized root system R in RN and a multiplicity function k≥0 let N=N+∑α∈Rk(α). Denote by dw(x)=∏α∈R 〈x,α〉 k(α)dx the associated measure in RN. Let F stand for the Dunkl transform. Given a bounded function m on RN, we prove that if there is s>N such that m satisfies the classical Hörmander condition with the smoothness s, then the multiplier operator Tmf=F−1(mFf) is of weak type (1,1), strong type (p,p) for 1
Abstract: Publication date: Available online 20 March 2019Source: Journal of Functional AnalysisAuthor(s): Javier Alejandro Chávez-Domínguez, Verónica Dimant, Daniel Galicer We introduce the class of operator p-compact mappings and completely right p-nuclear operators, which are natural extensions to the operator space framework of their corresponding Banach operator ideals. We relate these two classes, define natural operator space structures and study several properties of these ideals. We show that the class of operator ∞-compact mappings in fact coincides with a notion already introduced by Webster in the nineties (in a very different language). This allows us to provide an operator space structure to Webster's class.
Abstract: Publication date: Available online 13 March 2019Source: Journal of Functional AnalysisAuthor(s): Guowei Dai The uniqueness obtained in [1] may not right if f(s)/sk is only decreasing for s>0. We correct this error by requiring f(s)/sk is “strictly” decreasing for s>0.
Abstract: Publication date: Available online 12 March 2019Source: Journal of Functional AnalysisAuthor(s): Haïm Brezis, Petru Mironescu We investigate the validity of the fractional Gagliardo-Nirenberg-Sobolev inequality(1)‖f‖Wr,q(Ω)≲‖f‖Ws1,p1(Ω)θ‖f‖Ws2,p2(Ω)1−θ,∀f∈Ws1,p1(Ω)∩Ws2,p2(Ω).Here, s1,s2,r are non-negative numbers (not necessarily integers), 1≤p1,p2,q≤∞, and we assume, for some θ∈(0,1), the standard relations(2)r
Abstract: Publication date: Available online 12 March 2019Source: Journal of Functional AnalysisAuthor(s): Alexandre Afgoustidis Alain Connes and Nigel Higson pointed out in the 1990s that the Connes-Kasparov “conjecture” for the K-theory of reduced group C⁎-algebras seemed, in the case of reductive Lie groups, to be a cohomological echo of a conjecture of George Mackey concerning the rigidity of representation theory along the deformation from a real reductive group to its Cartan motion group. For complex semisimple groups, Nigel Higson established in 2008 that Mackey's analogy is a real phenomenon, and does lead to a simple proof of the Connes-Kasparov isomorphism. We here turn to more general reductive groups and use our recent work on Mackey's proposal, together with Higson's work, to obtain a new proof of the Connes-Kasparov isomorphism.
Abstract: Publication date: Available online 12 March 2019Source: Journal of Functional AnalysisAuthor(s): Teun D.H. van Nuland We introduce a novel commutative C*-algebra CR(X) of functions on a symplectic vector space (X,σ) admitting a complex structure, along with a strict deformation quantization that maps a dense subalgebra of CR(X) to the resolvent algebra introduced by Buchholz and Grundling [2]. The associated quantization map is a field-theoretical Weyl quantization compatible with the work of Binz, Honegger and Rieckers [1]. We also define a Berezin-type quantization map on all of CR(X), which continuously and bijectively maps it onto the resolvent algebra.The commutative resolvent algebra CR(X), generally defined on a real inner product space X, intimately depends on the finite dimensional subspaces of X. We thoroughly analyze the structure of this algebra in the finite dimensional case by giving a characterization of its elements and by computing its Gelfand spectrum.
Abstract: Publication date: Available online 8 March 2019Source: Journal of Functional AnalysisAuthor(s): Van Hoang Nguyen Using a dimension reduction argument and a stability version of the weighted Sobolev inequality on half space recently proved by Seuffert [49], we establish, in this paper, some stability estimates (or quantitative estimates) for a family of the sharp Gagliardo–Nirenberg–Sobolev inequalities due to Del Pino and Dolbeault [19].
Abstract: Publication date: Available online 7 March 2019Source: Journal of Functional AnalysisAuthor(s): Luigi De Pascale, Jean Louet The Monge-Kantorovich problem for the W∞ distance presents several peculiarities. Among them the lack of convexity and then of a direct duality. We study in dimension 1 the dual problem introduced by Barron, Bocea and Jensen in [2]. We construct a couple of Kantorovich potentials which is non trivial in the best possible way. More precisely, we build a potential which is non constant around any point that the restrictable, minimizing plan moves at maximal distance. As an application, we show that the set of points which are displaced at maximal distance by a “locally optimal” transport plan is shared by all the other optimal transport plans, and we describe the general structure of all the one-dimensional optimal transport plans.
Abstract: Publication date: Available online 6 March 2019Source: Journal of Functional AnalysisAuthor(s): Judit Abardia-Evéquoz, Károly J. Böröczky, Mátyás Domokos, Dávid Kertész The space of continuous, SL(m,C)-equivariant, m≥2, and translation covariant valuations taking values in the space of real symmetric tensors on Cm≅R2m of rank r≥0 is completely described. The classification involves the moment tensor valuation for r≥1 and is analogous to the known classification of the corresponding tensor valuations that are SL(2m,R)-equivariant, although the method of proof cannot be adapted.
Abstract: Publication date: Available online 25 February 2019Source: Journal of Functional AnalysisAuthor(s): Jia-Feng Cao, Yihong Du, Fang Li, Wan-Tong Li We introduce and study a class of free boundary models with “nonlocal diffusion”, which are natural extensions of the free boundary models in [16] and elsewhere, where “local diffusion” is used to describe the population dispersal, with the free boundary representing the spreading front of the species. We show that this nonlocal problem has a unique solution defined for all time, and then examine its long-time dynamical behavior when the growth function is of Fisher-KPP type. We prove that a spreading-vanishing dichotomy holds, though for the spreading-vanishing criteria significant differences arise from the well known local diffusion model in [16].
Abstract: Publication date: Available online 15 February 2019Source: Journal of Functional AnalysisAuthor(s): Daniela Kraus, Oliver Roth, Matthias Schötz, Stefan Waldmann On the Poincaré disc and its higher-dimensional analogs one has a canonical formal star product of Wick type. We define a locally convex topology on a certain class of real-analytic functions on the disc for which the star product is continuous and converges as a series. The resulting Fréchet algebra is characterized explicitly in terms of the set of all holomorphic functions on an extended and doubled disc of twice the dimension endowed with the natural topology of locally uniform convergence. We discuss the holomorphic dependence on the deformation parameter and the positive functionals and their GNS representations of the resulting Fréchet algebra.
Abstract: Publication date: Available online 14 February 2019Source: Journal of Functional AnalysisAuthor(s): Rafael Correa, Abderrahim Hantoute, Pedro Pérez-Aros This work provides formulae for the ε-subdifferential of integral functions in the framework of complete σ-finite measure spaces and locally convex spaces. In this work we present here new formulae for this ε-subdifferential under the presence of continuity-type qualification conditions relying on the data involved in the integrand.
Abstract: Publication date: Available online 14 February 2019Source: Journal of Functional AnalysisAuthor(s): N.V. Krylov We prove that, if s∉{1,2,...}, then any Ck(B¯1)-function can be approximated in the norm of this space by the functions which are s-harmonic in B1.
Abstract: Publication date: Available online 13 February 2019Source: Journal of Functional AnalysisAuthor(s): Arnaud Brothier, Vaughan F.R. Jones We introduce the Pythagorean C*-algebras and use the category/functor method to construct unitary representations of Thompson's groups from representations of them. We calculate several examples.
Abstract: Publication date: Available online 13 February 2019Source: Journal of Functional AnalysisAuthor(s): Youngae Lee, Chang-Shou Lin The seminal work [7] by Brezis and Merle showed that the bubbling solutions of the mean field equation have the property of mass concentration. Recently, Lin and Tarantello in [30] found that the “bubbling implies mass concentration” phenomena might not hold if there is a collapse of singularities. Furthermore, a sharp estimate [23] for the bubbling solutions has been obtained. In this paper, we prove that there exists at most one sequence of bubbling solutions if the collapsing singularity occurs. The main difficulty comes from that after re-scaling, the difference of two solutions locally converges to an element in the kernel space of the linearized operator. It is well-known that the kernel space is three dimensional. So the main technical ingredient of the proof is to show that the limit after re-scaling is orthogonal to the kernel space.
Abstract: Publication date: Available online 13 February 2019Source: Journal of Functional AnalysisAuthor(s): Yong Huang, Yongsheng Jiang In this paper, we give a variational analysis to the planar dual Minkowski problem in the Sobolev space. With the new variational characterization, we can deal with existence results for prescribed not necessarily positive data. Meanwhile, functional inequalities and multiple solutions are also obtained.
Abstract: Publication date: Available online 12 February 2019Source: Journal of Functional AnalysisAuthor(s): Leandro Candido, Marek Cúth, Michal Doucha We develop tools for proving isomorphisms of normed spaces of Lipschitz functions over various doubling metric spaces and Banach spaces. In particular, we show that Lip0(Zd)≃Lip0(Rd), for all d∈N. More generally, we e.g. show that Lip0(Γ)≃Lip0(G), where Γ is from a large class of finitely generated nilpotent groups and G is its Mal'cev closure; or that Lip0(ℓp)≃Lip0(Lp), for all 1≤p
Abstract: Publication date: Available online 7 February 2019Source: Journal of Functional AnalysisAuthor(s): Hervé Oyono-Oyono, Guoliang Yu In this paper, we apply quantitative operator K-theory to develop an algorithm for computing K-theory for the class of filtered C⁎-algebras with asymptotic finite nuclear decomposition. As a consequence, we prove the Künneth formula for C⁎-algebras in this class. Our main technical tool is a quantitative Mayer–Vietoris sequence for K-theory of filtered C⁎-algebras.
Abstract: Publication date: Available online 1 February 2019Source: Journal of Functional AnalysisAuthor(s): Giovanni Leoni, Ian Tice In this paper we construct a trace operator for homogeneous Sobolev spaces defined on infinite strip-like domains. We identify an intrinsic seminorm on the resulting trace space that makes the trace operator bounded and allows us to construct a bounded right inverse. The intrinsic seminorm involves two features not encountered in the trace theory of bounded Lipschitz domains or half-spaces. First, due to the strip-like structure of the domain, the boundary splits into two infinite disconnected components. The traces onto each component are not completely independent, and the intrinsic seminorm contains a term that measures the difference between the two traces. Second, as in the usual trace theory, there is a term in the seminorm measuring the fractional Sobolev regularity of the trace functions with a difference quotient integral. However, the finite width of the strip-like domain gives rise to a screening effect that bounds the range of the difference quotient perturbation. The screened homogeneous fractional Sobolev spaces defined by this screened seminorm on arbitrary open sets are of independent interest, and we study their basic properties. We conclude the paper with applications of the trace theory to partial differential equations.
Abstract: Publication date: Available online 15 January 2019Source: Journal of Functional AnalysisAuthor(s): Yurii Belov, Alexander Borichev, Konstantin Fedorovskiy Nevanlinna domains are an important class of bounded simply connected domains in the complex plane; they are images of the unit disc under mappings by univalent functions belonging to model spaces (i.e. the subspaces of the Hardy space H2 invariant with respect to the backward shift operator). Nevanlinna domains play a crucial role in recent progress in problems of uniform approximation of functions on compact sets in C by polyanalytic polynomials and polyanalytic rational functions. We give a complete solution to the following problem posed in the early 2000s: how large (in the sense of dimension) can be the boundaries of Nevanlinna domains' We establish the existence of Nevanlinna domains with large boundaries. In particular, these domains can have boundaries of positive planar measure. The sets of accessible points can be of any Hausdorff dimension between 1 and 2. As a quantitative counterpart of these results, we construct rational functions univalent in the unit disc with extremely long boundaries for a given amount of poles.
Abstract: Publication date: Available online 11 January 2019Source: Journal of Functional AnalysisAuthor(s): Raphaël Clouâtre, Christopher Ramsey We study non-selfadjoint operator algebras that can be entirely understood via their finite-dimensional representations. In contrast with the elementary matricial description of finite-dimensional C⁎-algebras, in the non-selfadjoint setting we show that an additional level of flexibility must be allowed. Motivated by this peculiarity, we consider a natural non-selfadjoint notion of residual finite-dimensionality. We identify sufficient conditions for the tensor algebra of a C⁎-correspondence to enjoy this property. To clarify the connection with the usual self-adjoint notion, we investigate the residual finite-dimensionality of the minimal and maximal C⁎-covers associated to an operator algebra.
Abstract: Publication date: Available online 11 January 2019Source: Journal of Functional AnalysisAuthor(s): Vaios Laschos, Alexander Mielke We study general geometric properties of cone spaces, and we apply them on the Hellinger–Kantorovich space (M(X),HKα,β). We exploit a two-parameter scaling property of the Hellinger–Kantorovich metric HKα,β, and we prove the existence of a distance SHKα,β on the space of probability measures that turns the Hellinger–Kantorovich space (M(X),HKα,β) into a cone space over the space of probabilities measures (P(X),SHKα,β). We provide a two parameter rescaling of geodesics in (M(X),HKα,β), and for (P(X),SHKα,β) we obtain a full characterization of the geodesics. We finally prove finer geometric properties, including local-angle condition and partial K-semiconcavity of the squared distances, that will be used in a future paper to prove existence of gradient flows on both spaces.
Abstract: Publication date: Available online 9 January 2019Source: Journal of Functional AnalysisAuthor(s): Sérgio Almaraz, Olivaine S. de Queiroz, Shaodong Wang Let (M,g) be a compact Riemannian three-dimensional manifold with boundary. We prove the compactness of the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. This involves a blow-up analysis of a Yamabe-type equation with critical Sobolev exponent on the boundary.
Abstract: Publication date: Available online 9 January 2019Source: Journal of Functional AnalysisAuthor(s): Lu Chen, Guozhen Lu, Chunxia Tao In this paper, we establish the existence of extremals for two kinds of Stein–Weiss inequalities on the Heisenberg group. More precisely, we prove the existence of extremals for the Stein–Weiss inequalities with full weights in Theorem 1.1 and the Stein–Weiss inequalities with horizontal weights in Theorem 1.4. Different from the proof of the analogous inequality in Euclidean spaces given by Lieb [26] using Riesz rearrangement inequality which is not available on the Heisenberg group, we employ the concentration compactness principle to obtain the existence of the maximizers on the Heisenberg group. Our result is also new even in the Euclidean case because we don't assume that the exponents of the double weights in the Stein–Weiss inequality (1.1) are both nonnegative (see Theorem 1.3 and more generally Theorem 1.5). Therefore, we extend Lieb's celebrated result of the existence of extremal functions of the Stein–Weiss inequality in the Euclidean space to the case where the exponents are not necessarily both nonnegative (see Theorem 1.3). Furthermore, since the absence of translation invariance of the Stein–Weiss inequalities, additional difficulty presents and one cannot simply follow the same line of Lions' idea to obtain our desired result. Our methods can also be used to obtain the existence of optimizers for several other weighted integral inequalities (Theorem 1.5).
Abstract: Publication date: Available online 9 January 2019Source: Journal of Functional AnalysisAuthor(s): Alexandru Chirvasitu, Benjamin Passer When a compact quantum group H coacts freely on unital C⁎-algebras A and B, the existence of equivariant maps A→B may often be ruled out due to the incompatibility of some invariant. We examine the limitations of using invariants, both concretely and abstractly, to resolve the noncommutative Borsuk–Ulam conjectures of Baum–Dąbrowski–Hajac. Among our results, we find that for certain finite-dimensional H, there can be no well-behaved invariant which solves the Type 1 conjecture for all free coactions of H. This claim is in stark contrast to the case when H is finite-dimensional and abelian. In the same vein, it is possible for all iterated joins of H to be cleft as comodules over the Hopf algebra associated to H. Finally, two commonly used invariants, the local-triviality dimension and the spectral count, may both change in a θ-deformation procedure.
Abstract: Publication date: Available online 21 December 2018Source: Journal of Functional AnalysisAuthor(s): Pablo Turco, Román Villafañe We establish a relation between Lipschitz operator ideals and linear operator ideals, which fits in the framework of Galois connection between lattices. We use this relationship to give a criterion which allows us to recognize when a Banach Lipschitz operator ideal is of composition type or not. Also, we introduce the concept of minimal Banach Lipschitz operator ideal, which have analogous properties to minimal Banach operator ideals. We characterize minimal Banach Lipschitz operator ideals which are of composition type and present examples which are not of this class.
Abstract: Publication date: Available online 21 December 2018Source: Journal of Functional AnalysisAuthor(s): Bernardo Cascales, Rafael Chiclana, Luis C. García-Lirola, Miguel Martín, Abraham Rueda Zoca We study the set SNA(M,Y) of those Lipschitz maps from a (complete pointed) metric space M to a Banach space Y which (strongly) attain their Lipschitz norm (i.e. the supremum defining the Lipschitz norm is a maximum). Extending previous results, we prove that this set is not norm dense when M is a length space (or local) or when M is a closed subset of R with positive Lebesgue measure, providing new examples which have very different topological properties than the previously known ones. On the other hand, we study the linear properties which are sufficient to get Lindenstrauss property A for the Lipschitz-free space F(M) over M, and show that all of them actually provide the norm density of SNA(M,Y) in the space of all Lipschitz maps from M to any Banach space Y. Next, we prove that SNA(M,R) is weakly sequentially dense in the space of all Lipschitz functions for all metric spaces M. Finally, we show that the norm of the bidual space of F(M) is octahedral provided the metric space M is discrete but not uniformly discrete or M′ is infinite.
Abstract: Publication date: Available online 21 December 2018Source: Journal of Functional AnalysisAuthor(s): Alain Connes, Galina Levitina, Edward McDonald, Fedor Sukochev, Dmitriy Zanin We introduce the notion of a pre-spectral triple, which is a generalisation of a spectral triple (A,H,D) where D is no longer required to be self-adjoint, but closed and symmetric. Despite having weaker assumptions, pre-spectral triples allow us to introduce noncompact noncommutative geometry with boundary. In particular, we derive the Hochschild character theorem in this setting. We give a detailed study of Dirac operators with Dirichlet boundary conditions on domains in Rd, d≥2.
Abstract: Publication date: Available online 21 December 2018Source: Journal of Functional AnalysisAuthor(s): Danko R. Jocić If X and Y are Banach spaces and f:BX→Y is Fréchet differentiable on the open unit ball BX of X, then for every operator monotone function φ:(−1,1)→R, which satisfies φ′′⩾0 on [a,b),(1)supa,b∈BX,a≠b‖f(a)−f(b)‖φ′(‖a‖)‖a−b‖φ′(‖b‖)=supa∈BX‖Df(a)‖φ′(‖a‖). This generalizes Holland–Walsh–Pavlović criterium for the membership in Bloch type spaces for functions defined in the unit ball of a Banach space and taking values in another Banach space. We also established relations of the induced Bloch and Lipschitz spaces with other spaces of vector valued functions.
Abstract: Publication date: Available online 17 December 2018Source: Journal of Functional AnalysisAuthor(s): A.S. Üstünel Let (W,H,μ) be the classical Wiener space on IRd. Assume that X=(Xt) is a diffusion process satisfying the stochastic differential equation dXt=σ(t,X)dBt+b(t,X)dt, where σ:[0,1]×C([0,1],IRn)→IRn⊗IRd, b:[0,1]×C([0,1],IRn)→IRn, B is an IRd-valued Brownian motion. We suppose that the weak uniqueness of this equation holds for any initial condition. We prove that any square integrable martingale M w.r.t. to the filtration (Ft(X),t∈[0,1]) can be represented asMt=E[M0]+∫0tPs(X)αs(X).dBs where α(X) is an IRd-valued process adapted to (Ft(X),t∈[0,1]), satisfying E∫0t(a(Xs)αs(X),α
Abstract: Publication date: Available online 5 December 2018Source: Journal of Functional AnalysisAuthor(s): Ryosuke Sato We propose a natural quantized character theory for inductive systems of compact quantum groups based on KMS states on AF-algebras following Stratila–Voiculescu's work [24] (or [8]), and give its serious investigation when the system consists of quantum unitary groups Uq(N) with q∈(0,1). The key features of this work are: The “quantized trace” of a unitary representation of a compact quantum group can be understood as a quantized character associated with the unitary representation and its normalized one is captured as a KMS state with respect to a certain one-parameter automorphism group related to the so-called scaling group. In this paper we provide a Vershik–Kerov type approximation theorem for extremal quantized characters (called the ergodic method) and also compare our quantized character theory for the inductive system of Uq(N) with Gorin's theory on q-Gelfand–Tsetlin graphs [11].
Abstract: Publication date: Available online 26 November 2018Source: Journal of Functional AnalysisAuthor(s): Ming-Liang Chen, Jing-Cheng Liu In this paper, we consider the planar self-affine measures μM,D generated by an expanding matrix M∈M2(Z) and an integer digit set D={(00),(α1α2),(β1β2)} with α1β2−α2β1≠0. We show that if det(M)∉3Z, then the mutually orthogonal exponential functions in L2(μM,D) is finite, and the exact maximal cardinality is given.
Abstract: Publication date: Available online 26 November 2018Source: Journal of Functional AnalysisAuthor(s): Minhyun Kim, Panki Kim, Jaehun Lee, Ki-Ahm Lee We study the boundary regularity of solutions of the Dirichlet problem for the nonlocal operator with a kernel of variable orders. Since the order of differentiability of the kernel is not represented by a single number, we consider the generalized Hölder space. We prove that there exists a unique viscosity solution of Lu=f in D, u=0 in Rn∖D, where D is a bounded C1,1 open set, and that the solution u satisfies u∈CV(D) and u/V(dD)∈Cα(D) with the uniform estimates, where V is the renewal function and dD(x)=dist(x,∂D).
Abstract: Publication date: Available online 20 November 2018Source: Journal of Functional AnalysisAuthor(s): Michael Cwikel, Amit Einav In this work we present a newly developed study of the interpolation of weighted Sobolev spaces by the complex method. We show that in some cases, one can obtain an analogue of the famous Stein–Weiss theorem for weighted Lp spaces. We consider an example which gives some indication that this may not be possible in all cases. Our results apply in cases which cannot be treated by methods in earlier papers about interpolation of weighted Sobolev spaces. They include, for example, a proof that [W1,p(Rd,ω0),W1,p(Rd,ω1)]θ=W1,p(Rd,ω01−θω1θ) whenever ω0 and ω1 are continuous and their quotient is the exponential of a Lipschitz function. We also mention some possible applications of such interpolation in the study of convergence in evolution equations.
Abstract: Publication date: Available online 20 November 2018Source: Journal of Functional AnalysisAuthor(s): David Damanik, Selim Sukhtaiev We prove Anderson localization for the discrete Laplace operator on radial tree graphs with random branching numbers. Our method relies on the representation of the Laplace operator as the direct sum of half-line Jacobi matrices whose entries are non-degenerate, independent, identically distributed random variables with singular distributions.
Abstract: Publication date: Available online 16 November 2018Source: Journal of Functional AnalysisAuthor(s): Shaowei Chen, Jiaquan Liu, Zhi-Qiang Wang For semiclassical nonlinear Schrödinger equations with Sobolev critical exponent we propose and implement a procedure to construct an unbounded sequence of semiclassical bound states which are localized nodal solutions and are concentrated at a local minimum set of the potential. Our approach does not need any non-degeneracy condition of the limiting equation and is robust for more general problems.
Abstract: Publication date: Available online 14 November 2018Source: Journal of Functional AnalysisAuthor(s): Guozheng Cheng, Xiang Fang, Sen Zhu In this paper we initiate the study of a fundamental yet untapped random model of non-selfadjoint, bounded linear operators acting on a separable complex Hilbert space. We replace the weights wn=1 in the classical unilateral shift T, defined as Ten=wnen+1, where {en}n=1∞ form an orthonormal basis of a complex Hilbert space, by a sequence of i.i.d. random variables {Xn}n=1∞; that is, wn=Xn. This paper answers basic questions concerning such a model. We propose that this model can be studied in comparison with the classical Hardy/Bergman/Dirichlet spaces in function-theoretic operator theory.We calculate the spectra and determine their fine structures (Section 3). We classify the samples up to four equivalence relationships (Section 4). We introduce a family of random Hardy spaces and determine the growth rate of the coefficients of analytic functions in these spaces (Section 5). We compare them with three types of classical operators (Section 6); this is achieved in the form of generalized von Neumann inequalities. The invariant subspaces are shown to admit arbitrarily large indices and their semi-invariant subspaces model arbitrary contractions almost surely. We discuss a Beurling-type theorem (Section 7). We determine various non-selfadjoint algebras generated by T (Section 8). Their dynamical properties are clarified (Section 9). Their iterated Aluthge transforms are shown to converge (Section 10).In summary, they provide a new random model from the viewpoint of probability theory, and they provide a new class of analytic functional Hilbert spaces from the viewpoint of operator theory. The technical novelty in this paper is that the methodology used draws from three (largely separate) sources: probability theory, functional Hilbert spaces, and the approximation theory of bounded operators.
Abstract: Publication date: Available online 13 November 2018Source: Journal of Functional AnalysisAuthor(s): Frédéric Bayart We prove the existence of algebras of hypercyclic vectors in three cases: convolution operators, composition operators, and backward shift operators.
Abstract: Publication date: Available online 9 November 2018Source: Journal of Functional AnalysisAuthor(s): Hun Hee Lee, Ebrahim Samei, Nico Spronk We address two errors made in our paper [7]. The most significant error is in Theorem 1.1. We repair this error, and show that the main result, Theorem 2.5 of [7], is true. The second error is in one of our examples, Remark 2.4 (iv), and we partially resolve it.
Abstract: Publication date: Available online 6 November 2018Source: Journal of Functional AnalysisAuthor(s): Biswarup Das, Matthew Daws, Pekka Salmi We give a partial solution to a long-standing open problem in the theory of quantum groups, namely we prove that all finite-dimensional representations of a wide class of locally compact quantum groups factor through matrix quantum groups (Admissibility Conjecture for quantum group representations). We use this to study Kazhdan's Property (T) for quantum groups with non-trivial scaling group, strengthening and generalising some of the earlier results obtained by Fima, Kyed and Sołtan, Chen and Ng, Daws, Skalski and Viselter, and Brannan and Kerr. Our main results are:(i)All finite-dimensional unitary representations of locally compact quantum groups which are either unimodular or arise through a special bicrossed product construction are admissible.(ii)A generalisation of a theorem of Wang which characterises Property (T) in terms of isolation of finite-dimensional irreducible representations in the spectrum.(iii)A very short proof of the fact that quantum groups with Property (T) are unimodular.(iv)A generalisation of a quantum version of a theorem of Bekka–Valette proven earlier for quantum groups with trivial scaling group, which characterises Property (T) in terms of non-existence of almost invariant vectors for weakly mixing representations.(v)A generalisation of a quantum version of Kerr–Pichot theorem, proven earlier for quantum groups with trivial scaling group, which characterises Property (T) in terms of denseness properties of weakly mixing representations.
Abstract: Publication date: Available online 6 November 2018Source: Journal of Functional AnalysisAuthor(s): Bartłomiej Dyda, Moritz Kassmann We study function spaces and extension results in relation with Dirichlet problems involving integrodifferential operators. For such problems, data are prescribed on the complement of a given domain Ω⊂Rd. We introduce a function space that serves as a trace space for nonlocal Dirichlet problems and study related extension results.
Abstract: Publication date: Available online 1 November 2018Source: Journal of Functional AnalysisAuthor(s): Vladimir Derkach, Harry Dym De Branges–Pontryagin spaces B(E) with negative index κ of entire p×1 vector valued functions based on an entire p×2p entire matrix valued function E(λ) (called the de Branges matrix) are studied. An explicit description of these spaces and an explicit formula for the indefinite inner product are presented. A characterization of those spaces B(E) that are invariant under the generalized backward shift operator that extends known results when κ=0 is given. The theory of rigged de Branges–Pontryagin spaces is developed and then applied to obtain an embedding of de Branges matrices with negative squares in generalized J-inner matrices and selfadjoint extensions of the multiplication operator in B(E). A formula for factoring an arbitrary generalized J-inner matrix valued function into the product of a singular factor and a perfect factor is found analogous to the known factorization formulas for J-inner matrix valued functions.
Abstract: Publication date: Available online 30 October 2018Source: Journal of Functional AnalysisAuthor(s): Li-Xiang An, Liu He, Xing-Gang He Let μ be a Borel probability measure with compact support in R. The μ is called a spectral/Riesz spectral/frame spectral measure if there exists a set Λ⊂R such that the family of exponential functionsEΛ={e2πiλx:λ∈Λ} forms an orthonormal basis/Riesz basis/frame for L2(μ). In this paper we study the spectrality and the non-spectrality of a class of singular measures, which is one of the fundamental works for the analysis on L2(μ). For a clear expression, we use the simplest setting in this paper although the most results of ours can be extended to more general cases, even to higher dimensions.
Abstract: Publication date: Available online 30 October 2018Source: Journal of Functional AnalysisAuthor(s): Mariusz Mirek, Elias M. Stein, Bartosz Trojan We prove ℓp(Zd) bounds, for p∈(1,∞), of discrete maximal functions corresponding to averaging operators and truncated singular integrals of Radon type, and their applications to pointwise ergodic theory. Our new approach is based on a unified analysis of both types of operators, and also yields an extension to the vector-valued form of these results.
Abstract: Publication date: Available online 30 October 2018Source: Journal of Functional AnalysisAuthor(s): Evgeny Abakumov, Anton Baranov, Yurii Belov We extend some results of M. G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an application we obtain new versions of de Branges' Ordering Theorem for nearly invariant subspaces in a class of Hilbert spaces of entire functions. Examples illustrating sharpness of the obtained results are given. We also discuss applications to spectral theory of rank one perturbations of normal operators.
Abstract: Publication date: Available online 27 October 2018Source: Journal of Functional AnalysisAuthor(s): Camila F. Sehnem Let P be a unital subsemigroup of a group G. We propose an approach to C⁎-algebras associated to product systems over P. We call the C⁎-algebra of a given product system E its covariance algebra and denote it by A×EP, where A is the coefficient C⁎-algebra. We prove that our construction does not depend on the embedding P↪G and that a representation of A×EP is faithful on the fixed-point algebra for the canonical coaction of G if and only if it is faithful on A. We compare this with other constructions in the setting of irreversible dynamical systems, such as Cuntz–Nica–Pimsner algebras, Fowler's Cuntz–Pimsner algebra, semigroup C⁎-algebras of Xin Li and Exel's crossed products by interaction groups.
Abstract: Publication date: Available online 19 October 2018Source: Journal of Functional AnalysisAuthor(s): Michiya Mori We investigate surjective isometries between projection lattices of two von Neumann algebras. We show that such a mapping is characterized by means of Jordan ⁎-isomorphisms. In particular, we prove that two von Neumann algebras without type I1 direct summands are Jordan ⁎-isomorphic if and only if their projection lattices are isometric. Our theorem extends the recent result for type I factors by G.P. Gehér and P. Šemrl, which is a generalization of Wigner's theorem.
Abstract: Publication date: Available online 16 October 2018Source: Journal of Functional AnalysisAuthor(s): Ruxi Shi Let P={pn}n=1∞ be a sequence of positive integers with pn>1 and D={Dn}n=1∞ be a sequence of subsets of N with CardDn=2 or 3. The infinite convolution of probability measuresμP,D=δp1−1D1⁎δ(p1p2)−1D2⁎⋯ is a Borel probability measure which is singular with respect to the Lebesgue measure on R. In this paper, we prove the spectrality of such measure, i.e., to find a set Λ⊂R such that the set {e−2πiλx λ∈Λ} is an orthonormal basis of L2(μP,D).
Abstract: Publication date: Available online 16 October 2018Source: Journal of Functional AnalysisAuthor(s): Stefan Richter, Faruk Yilmaz Let D denote the classical Dirichlet space of analytic functions on the open unit disc whose derivative is square area integrable. For a set E⊆∂D we writeDE={f∈D:limr→1−f(reit)=0q.e.}, where q.e. stands for “except possibly for eit in a set of logarithmic capacity 0”. We show that if E is a Carleson set, then there is a function f∈DE that is also in the disc algebra and that generates DE in the sense that DE=clos{pf:p is a polynomial}.We also show that if φ∈D is an extremal function (i.e. 〈pφ,φ〉=p(0) for every polynomial p), then the limits of φ(z) exist for every eit∈∂D as z approaches eit from within any polynomially tangential approach region.
Abstract: Publication date: Available online 12 October 2018Source: Journal of Functional AnalysisAuthor(s): Bachir Bekka Answering a question of J. Rosenberg from [21], we construct the first examples of infinite characters on GLn(K) for a global field K and n≥2. The case n=2 is deduced from the following more general result. Let G a non amenable countable group acting on a locally finite tree X. Assume either that the stabilizer in G of every vertex of X is finite or that the closure of the image of G in Aut(X) is not amenable. We show that G has uncountably many infinite dimensional irreducible unitary representations (π,H) of G which are traceable, that is, such that the C⁎-subalgebra of B(H) generated by π(G) contains the algebra of the compact operators on H. In the case n≥3, we prove the existence of infinitely many characters for G=GLn(R), when R is an integral domain such that G is not amenable. In particular, the group SLn(Z) has infinitely many such characters for n≥2.
Abstract: Publication date: Available online 27 August 2018Source: Journal of Functional AnalysisAuthor(s): Danko R. Jocić, Đorđe Krtinić, Milan Lazarević In this note we show that the presented proof of [1, th. 12], being based on a false statement appearing in this proof, is not viable for all of the proclaimed values of the involved parameters. We also determine necessary and sufficient conditions for those parameters, which provides that the considered statement, and therefore [1, th. 12] itself, still remains valid.
Abstract: Publication date: Available online 27 August 2018Source: Journal of Functional AnalysisAuthor(s): Hua Qiu For a certain domain Ω in the Sierpinski gasket SG whose boundary is a line segment, a complete description of the eigenvalues of the Laplacian, with an exact count of dimensions of eigenspaces, under the Dirichlet and Neumann boundary conditions is presented. The method developed in this paper is a weak version of the spectral decimation method due to Fukushima and Shima, since for a lot of “bad” eigenvalues the spectral decimation method can not be used directly. Let ρ0(x), ρΩ(x) be the eigenvalue counting functions of the Laplacian associated to SG and Ω respectively. We prove a comparison between ρ0(x) and ρΩ(x) says that 0≤ρ0(x)−ρΩ(x)≤Cxlog2/log5logx for sufficiently large x for some positive constant C. As a consequence, ρΩ(x)=g(logx)xlog3/log5+O(xlog2/log5logx) as x→∞, for some (right-continuous discontinuous) log5-periodic function g:R→R with 0
Abstract: Publication date: Available online 24 August 2018Source: Journal of Functional AnalysisAuthor(s): Mathew Gluck In this work we obtain sharp embedding inequalities for a family of conformally invariant integral extension operators. This family includes (among others) the classical Poisson extension operator and the extension operator with Riesz kernel. We show that the sharp constants in these inequalities are attained and classify the corresponding extremal functions. We also compute the limiting behavior at the boundary of the extensions of the extremal functions.
Abstract: Publication date: Available online 24 August 2018Source: Journal of Functional AnalysisAuthor(s): Alexandru Aleman, Bartosz Malman We consider Hilbert spaces of analytic functions in the disk with a normalized reproducing kernel and such that the backward shift f(z)↦f(z)−f(0)z is a contraction on the space. We present a model for this operator and use it to prove the surprising result that functions which extend continuously to the closure of the disk are dense in the space. This has several applications, for example we can answer a question regarding reverse Carleson embeddings for these spaces. We also identify a large class of spaces which are similar to the de Branges–Rovnyak spaces and prove some results which are new even in the classical case.
Abstract: Publication date: Available online 3 July 2018Source: Journal of Functional AnalysisAuthor(s): Helmut Maier, Michael Th. Rassias We give an estimate for sums appearing in the Nyman–Beurling criterion for the Riemann Hypothesis. These sums contain the Möbius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably sharp in comparison to other sums containing the Möbius function. The bound is smaller than the trivial bound – essentially the number of terms – by a fixed power of that number. The exponent is made explicit. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.
Abstract: Publication date: Available online 7 June 2018Source: Journal of Functional AnalysisAuthor(s): Jingbo Dou, Meijun Zhu In this paper we introduce and study some nonlinear integral equations on bounded domains that are related to the sharp Hardy–Littlewood–Sobolev inequality. Existence results as well as nonexistence results are obtained.
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