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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 FROM AUTHOR]

Abstract: We compare the above-mentioned article with the content of a previous publication. [Copyright &y& Elsevier]

In the first part of the paper, we establish continuous bilinear decompositions that arise in the study of products between elements in martingale Hardy spaces H p (0 < p ⩽ 1) and functions in their dual spaces. Our decompositions are based on martingale paraproducts. The second part of the paper concerns applications of the method developed for the classical martingales in the first part. In particular, we build a connection between Hardy spaces on spaces of homogenous type equipped with a doubling measure and Hardy spaces with respect to the corresponding dyadic martingales. Using the method introduced in the first part, we obtain analogous results for dyadic martingales on spaces of homogenous type, which, thanks to the aforementioned connection, yield conclusions for products between elements in Hardy spaces and those in their duals on spaces of homogenous type. The key property of martingale Hardy spaces in the study is that they admit the atomic decomposition, for which we provide an interpretation via duality. [ABSTRACT FROM AUTHOR]

The basic disentanglement theorem established by the present authors states that estimates on a weighted geometric mean over (convex) families of functions can be disentangled into quantitatively linked estimates on each family separately. On the one hand, the theorem gives a uniform approach to classical results including Maurey's factorisation theorem and Lozanovskiĭ's factorisation theorem, and, on the other hand, it underpins the duality theory for multilinear norm inequalities developed in our previous two papers. In this paper we give a simple proof of this basic disentanglement theorem. Whereas the approach of our previous paper was rather involved – it relied on the use of minimax theory together with weak*-compactness arguments in the space of finitely additive measures, and an application of the Yosida–Hewitt theory of such measures – the alternate approach of this paper is rather straightforward: it instead depends upon elementary perturbation and compactness arguments. [ABSTRACT FROM AUTHOR]

We first give a precise description of an error in our paper "A T 1 theorem for general Calderón-Zygmund operators with doubling weights, and optimal cancellation conditions, II" published in JFA [1]. We then provide an overview of the correction needed, pointing to details appearing in the corrected (and retitled) paper "A weak to strong type T 1 theorem for general smooth Calderón-Zygmund operators with doubling weights, II" in arXiv:2111.06277v5 [2]. [ABSTRACT FROM AUTHOR]

In this paper, we will compute the Morse index and nullity of some embedded stationary geodesic networks in the sphere. The key theorem in the computation is that the index (and nullity) for the whole network is related to the index (and nullity) of small networks and the Dirichlet-to-Neumann map defined in this paper. Finally, we will show that for all stationary triple junction networks in S 2 , there is only one eigenvalue (without multiplicity) −1, which is less than 0, and the corresponding eigenfunctions are locally constant. Besides, the multiplicity of eigenvalues 0 is 3 for these networks, and their eigenfunctions are generated by the rotations on the sphere. [ABSTRACT FROM AUTHOR]

In the present paper we identify those filtered probability spaces (Ω , F , (F n) , P) that determine already the martingale type of a Banach space X. We isolate intrinsic conditions on the filtration (F n) of purely atomic σ -algebras which determine that the upper ℓ p estimates ‖ f ‖ L p (Ω , X) p ⩽ C p (‖ E (f | F 0) ‖ L p (Ω , X) p + ∑ n = 1 ∞ ‖ E (f | F n) − E (f | F n − 1) ‖ L p (Ω , X) p) , f ∈ L p (Ω , X) imply that the Banach space X is of martingale type p. Our paper complements G. Pisier's investigation [12] and continues the work by S. Geiss and second named author in [3]. [ABSTRACT FROM AUTHOR]

This paper aims to address the issues of the global existence and the large-time decay estimates of strong solutions for the two-dimensional compressible Prandtl equations with small initial data, which is analytical in the tangential variable. We investigate a more complicated system, which contains more physics than the incompressible system. Not only the loss of the horizontal derivative in the estimate of nonlinear terms, but also the failure to satisfy the divergence-free condition, may create significant difficulties when closing energy estimates. Motivated by Paicu and Zhang (2021) [36] for the incompressible case, we find a new quantity G = def u + y 2 〈 t 〉 φ and derive a sufficiently fast decay-in-time estimate of some weighted analytic norm to G. Together with a linear combination of the tangential velocity u with its primitive one φ , which controls the evolution of the analytic radius to solutions ultimately. Another interesting feature of this paper is to take into account the influence of the density, however, the density is independent of the y variable. In order to avoid these obstacles, the approaches we take are some delicate treatments, including the special cut-off function and new inequalities in the process of energy estimates. This paper can be considered as a first global-in-time Cauchy–Kowalevsakya result for the compressible Prandtl equations with small analytic data. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate a new locally constrained mean curvature flow (1.5) and prove that if the initial hypersurface Σ 0 is of smoothly closed starshaped, then the solution Σ t of the flow (1.5) exists for all time and converges to a sphere in C ∞ -topology. Following this flow argument, we obtain a new proof of the celebrated sharp Michael-Simon inequality (1.2) for mean curvatures on smooth, compact, and starshaped hypersurface M 0 (possibly with boundary). Specifically, if M 0 is closed, we find a necessary and sufficient condition for the equality of (1.2 ′) holds. In the second part of this paper, by exploiting the properties of the inverse mean curvature type flow (1.6) obtained from the expanding geometric flow in [7] by rescaling, we develop and present a new sharp Michael-Simon inequality (1.7) for the k -th mean curvatures. When M 0 is closed, starshaped, (k − 1)-convex and g is of constant, the inequality (1.7) is equivalent to the Alexandrov-Fenchel type inequality (1.8). [ABSTRACT FROM AUTHOR]

The author of this paper and coauthors in 2022 studied a family of self-affine tiles in R d with noncollinear digit sets, and gave a sufficient and necessary condition for such tiles to be tame balls. We in this paper mainly present a simpler proof of such equivalent condition. We replace quadric surfaces by some zigzag planes, and redefine the quasi-invariant plane which plays a key role in the construction of the desired homeomorphism. This adjustment greatly simplifies the proof. [ABSTRACT FROM AUTHOR]

Abstract: There are two main results in the paper. In the first one, Theorem 1, we prove that if the Sobolev embedding theorem holds in Ω, in any of all the possible cases, then Ω satisfies the measure density condition. The second main result, Theorem 5, provides several characterizations of the -extension domains for . As a corollary we prove that the property of being a -extension domain, , is invariant under bi-Lipschitz mappings, Theorem 8. [Copyright &y& Elsevier]

In this paper, we show that two Zariski dense subgroups consisting of hyperbolic elements in SO(n+1,n)⋉R2n+1 with the same marked Margulis invariant, are conjugate. We also consider in affine deformations an analogue of quasifuchsian deformation of Fuchsian groups. [Copyright &y& Elsevier]

This paper will focus on the proof of local integrability of Bessel functions for GL (n) (p -adic case) by using the relations between Bessel functions and local Kloosterman (orbital) integrals proved in several papers of E. M. Baruch [5] [6] [7] , the theory of the (relative) Shalika germs established by H. Jacquet and Y. Ye in [25] [26] and G. Stevens' approach [35] on estimating certain GL (n) generalized Kloosterman sums. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following superlinear elliptic problem (P) { − Δ u = λ | u | p − 2 u + f (x , u) , in Ω , u = 0 , on ∂ Ω , where λ > 0 and 2 < p < 2 + δ for some δ > 0 small. The nonlinearity f satisfies the Ambrosetti-Rabinowitz condition and other appropriate hypotheses such that u = 0 is a local minimizer of the associated energy functional of equation (P). Our main novelties are threefold. Firstly, using the properties of Gromoll-Meyer pairs in Morse theory, we prove that equation (P) has at least one nontrivial solution close to 0. Moreover, four nontrivial solutions are obtained with assumptions on f at infinity, and none of these solutions depends on the gaps of consecutive eigenvalues of operator −Δ. Therefore, our results differ significantly from those of the paper by Li and Li (2016) [16]. Secondly, under the assumptions of the paper above, we can obtain the existence of a fifth nontrivial solution of equation (P) for λ = 1. Finally, by using minimax methods and Morse theory, we also obtain the existence of five nontrivial solutions of equation (P) based on the relationship between parameter λ and eigenvalues of operator −Δ. [ABSTRACT FROM AUTHOR]

The signature of a path, introduced by K.T. Chen [10] in 1954, has been extensively studied in recent years. The fundamental 2010 paper [20] of Hambly and Lyons showed that the signature is an injective function on the space of continuous, finite-variation paths up to a general notion of reparameterisation called tree-like equivalence. This result has been extended to geometric rough paths by Boedihardjo et al. [5]. More recently, the approximation theory of the signature has been widely used in the literature in applications. The archetypal instance of these results, see e.g. [24] , guarantees uniform approximation, on compact sets, of a continuous function by a linear functional on the (extended) tensor algebra acting on the signature. In this paper we study in detail, and for the first time, the properties of three natural candidate topologies on the set of unparameterised paths, i.e. the tree-like equivalence classes. These are obtained by privileging different properties of the signature and are: (1) the product topology, obtained by equipping the range of the signature with the (subspace topology of the) product topology in the extended tensor algebra and then requiring S to be an embedding, (2) the quotient topology derived from the 1-variation topology on the underlying path space, and (3) the metric topology associated to d ([ γ ] , [ σ ]) : = | | γ ⁎ − σ ⁎ | | 1 using the (constant-speed) tree-reduced representatives γ ⁎ and σ ⁎ of the respective equivalence classes. We evaluate these spaces from the point of view of their suitability when it comes to studying (probability) measures on them. We prove that the respective collections of open sets are ordered by strict inclusion, (1) being the weakest and (3) the strongest. Our other conclusions can be summarised as follows. All three topological spaces are separable and Hausdorff, (1) being both metrisable and σ -compact, but not a Baire space and hence being neither Polish nor locally compact. The completion of (1), in any metric inducing the product topology, is the subspace G ⁎ of group-like elements. The quotient topology (2) is not metrisable and the metric d is not complete. We also discuss some open problems related to these spaces. We consider finally the implications of the selection of the topology for uniform approximation results involving the signature. A stereotypical model for a continuous function on (unparameterised) path space is the solution of a controlled differential equation. We thus prove, for a broad class of these equations, well-definedness and measurability of the (fixed-time) solution map with respect to the Borel sigma-algebra of each topology. Under stronger regularity assumptions, we further show continuity of this same map on explicit compact subsets of the product topology (1). We relate these results to the expected signature model of [24]. [ABSTRACT FROM AUTHOR]

This paper studies the existence and uniqueness problem for the generalized Benjamin-Bona-Mahony (gBBM) equation with quasi-periodic initial data on the real line. We obtain an existence and uniqueness result in the classical sense with arbitrary time horizon under the assumption of polynomially decaying initial Fourier data using the combinatorial analysis method developed in earlier papers by Christ [6] , Damanik-Goldstein [11] , and the present authors [12]. Our result is valid for exponentially decaying initial Fourier data and hence can be viewed as a Cauchy-Kovalevskaya theorem in the space variable for the gBBM equation with quasi-periodic initial data. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to find pointwise representation formulas for the density of the pairing between divergence-measure fields and BV functions, in this way continuing the research started in [17,20]. In particular, we extend a representation formula from an unpublished paper of Anzellotti [7] involving the limit of cylindrical averages for normal traces, and we exploit a result of [35] in order to derive another representation in terms of limits of averages in half balls. [ABSTRACT FROM AUTHOR]

In this paper, we attempt to resolve the singularities of the zero variety of a C ∞ function of two variables as much as possible by using ordinary blowings up. As a result, we formulate an algorithm to locally express the zero variety in the "almost" normal crossings form, which is close to the normal crossings form but may include flat functions. As an application, we investigate analytic continuation of local zeta functions associated with C ∞ functions of two variables. As is well known, the desingularization theorem of Hironaka implies that the local zeta functions associated with real analytic functions admit the meromorphic continuation to the whole complex plane. On the other hand, it is recently observed that the local zeta function associated with a specific (non-real analytic) C ∞ function has a singularity different from the pole. From this observation, the following questions are naturally raised in the C ∞ case: how wide the meromorphically extendible region can be and what kinds of information essentially determine this region? This paper shows that this region can be described in terms of some kind of multiplicity of the zero variety of each C ∞ function. By using our blowings up algorithm, it suffices to investigate local zeta functions in the almost normal crossings case. This case can be effectively analyzed by using real analysis methods; in particular, a van der Corput-type lemma plays a crucial role in the determination of the above region. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the mapping properties of pseudo-differential operators with operator-valued symbols. We prove the boundedness of regular symbols on Sobolev spaces H 2 α (R d ; L 2 (M)) and Besov spaces B p , q α (R d ; L p (M)) for α ∈ R and 1 ≤ p , q ≤ ∞ , as well as the boundedness of forbidden symbols on H 2 α (R d ; L 2 (M)) and B p , q α (R d ; L p (M)) for α > 0 and 1 ≤ p , q ≤ ∞. Thanks to the smooth atomic decomposition of the operator-valued Triebel-Lizorkin spaces F 1 α , c (R d , M) obtained in our previous paper, we establish the F 1 α , c -regularity of regular symbols for every α ∈ R , and the F 1 α , c -regularity of forbidden symbols for α > 0. As applications, we obtain the same results on the usual and quantum tori. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

In the paper, we consider a path-dependent Hamilton–Jacobi equation with coinvariant derivatives over the space of continuous functions. Such equations arise from optimal control problems and differential games for time-delay systems. We study generalized solutions of the considered Hamilton–Jacobi equation both in the minimax and in the viscosity sense. A minimax solution is defined as a functional which epigraph and subgraph satisfy certain conditions of weak invariance, while a viscosity solution is defined in terms of a pair of inequalities for coinvariant sub- and supergradients. We prove that these two notions are equivalent, which is the main result of the paper. As a corollary, we obtain comparison and uniqueness results for viscosity solutions of a Cauchy problem for the considered Hamilton–Jacobi equation and a right-end boundary condition. The proof of the main result is based on a certain property of the coinvariant subdifferential. To establish this property, we develop a technique going back to the proofs of multidirectional mean-value inequalities. In particular, the absence of the local compactness property of the underlying continuous function space is overcome by using Borwein–Preiss variational principle with an appropriate gauge-type functional. [ABSTRACT FROM AUTHOR]

In this paper we consider the gradient flow of the following Ginzburg-Landau type energy F ε (u) : = 1 2 ∫ M | D u | g 2 + 1 2 ε 2 (| u | g 2 − 1) 2 vol g. This energy is defined on tangent vector fields on a 2-dimensional closed and oriented Riemannian manifold M (here D stands for the covariant derivative) and depends on a small parameter ε > 0. If the energy satisfies proper bounds, when ε → 0 the second term forces the vector fields to have unit length. However, due to the incompatibility for vector fields on M between the Sobolev regularity and the unit norm constraint, critical points of F ε tend to generate a finite number of singular points (called vortices) having non-zero index (when the Euler characteristic is non-zero). These types of problems have been extensively analyzed in the recent paper by R. Ignat & R. Jerrard [19]. As in Euclidean case (see, among the others [8]), the position of the vortices is ruled by the so-called renormalized energy. In this paper we are interested in the dynamics of vortices. We rigorously prove that the vortices move according to the gradient flow of the renormalized energy, which is the limit behaviour when ε → 0 of the gradient flow of the Ginzburg-Landau energy. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dual fractional parabolic problem { ∂ t α u (x , t) + (− Δ) s u (x , t) = f (u (x , t)) , in R + n × R , u (x , t) = 0 , in (R n ﹨ R + n) × R , where R + n : = { x ∈ R n | x 1 > 0 } is the right half space. We prove that the positive solutions are strictly increasing in x 1 direction without assuming the solutions be bounded. So far as we know, this is the first paper to explore the monotonicity of possibly unbounded solutions for the nonlocal parabolic problem involving both the fractional time derivative ∂ t α and the fractional Laplacian (− Δ) s. To overcome the difficulties caused by the dual nonlocality in space-time and by the remarkably weak assumptions on solutions, we introduced several new ideas and our approaches are quite different from those in the previous literature. We first establish an unbounded narrow region principle without imposing any decay and boundedness assumptions on the antisymmetric functions at infinity by estimating the nonlocal operator ∂ t α + (− Δ) s along a sequence of suitable auxiliary functions at their minimum points, which is an essential ingredient to carry out the method of moving planes at the starting point. Then in order to remove the decay or boundedness assumption on the solutions, we develop a new novel approach lies in establishing the averaging effects for such nonlocal operator and apply these averaging effects twice to guarantee that the plane can be moved all the way to infinity to derive the monotonicity of solutions. We believe that the new ideas and techniques developed here will become very useful tools in studying the qualitative properties of solutions, in particular of those unbounded solutions, for a wide range of fractional elliptic and parabolic problems. [ABSTRACT FROM AUTHOR]

In this paper we provide an optimal estimate for the operator norm of time-frequency localization operators L F , φ : L 2 (R d) → L 2 (R d) , with Gaussian window φ and weight F , under the assumption that F ∈ L p (R 2 d) ∩ L q (R 2 d) for some p and q in (1 , + ∞). We are also able to characterize optimal weight functions, whose shape turns out to depend on the ratio ‖ F ‖ q / ‖ F ‖ p. Roughly speaking, if this ratio is "sufficiently large" or "sufficiently small" optimal weight functions are certain Gaussians, while if it is in the intermediate regime the optimal functions are no longer Gaussians. As an application, we extend Lieb's uncertainty inequality to the space L p + L q. [ABSTRACT FROM AUTHOR]

This is a corrigendum to the article: "Computation of maximal projection constants" (J. Funct. Anal., 277). The statement of Lemma 3.1(2) of that paper is incorrect. As a consequence of this the proof of Theorem 1.4 is incomplete. In this corrigendum we prove a corrected version of Lemma 3.1 and explain why, with this weaker result, our original strategy for proving Theorem 1.4 no longer works. The other results of the article, in particular the alternative proof of Grünbaum's conjecture, do not depend on Lemma 3.1 and are thus not affected by this error. [ABSTRACT FROM AUTHOR]

In this paper, we study the smallest gaps between successive zeros of nondegenerate smooth stationary centered Gaussian processes on the real line with the assumption that the covariance kernel κ (x) and its derivatives decay to 0 as | x | → ∞. We prove that, after rescaling, the smallest gaps converge to a Poisson point process with a specific rate. Moreover, the positions where these smallest gaps occur tend to a uniform distribution. Consequently, we can derive the limiting density for the k -th smallest gap. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the defocusing cubic nonlinear Schrödinger equation (NLS) on the waveguide manifold R 3 × T and establish almost sure scattering for random initial data, where no symmetry conditions are imposed and the result is available for arbitrarily rough data f ∈ H s with s ∈ R. The main new ingredient is a layer-by-layer refinement of the newly established randomization introduced by Shen-Soffer-Wu [40] , which enables us to also obtain a strongly smoothing effect from the randomization for the forcing term along the periodic direction. It is worth noting that such a smoothing effect generally can not hold for purely compact manifolds, which is on the contrary available for the present model thanks to the mixed type nature of the underlying domain. As a byproduct, by assuming that the initial data are periodically trivial, we also obtain almost sure scattering for the defocusing cubic NLS on R 3 which parallels the result by Camps [15] and Shen-Soffer-Wu [41]. To our knowledge, the paper also gives the first almost sure well-posedness and scattering result for NLS on product spaces. [ABSTRACT FROM AUTHOR]

In this paper, we apply various methods to establish the uniqueness of solutions to some classes of anisotropic and isotropic curvature problems. Firstly, by employing integral formulas derived by S. S. Chern [20] , we obtain the uniqueness of smooth admissible solutions to a class of Orlicz-(Christoffel)-Minkowski problems. Secondly, inspired by Simon's uniqueness result [61] , we then prove that the only smooth strictly convex solution to the following isotropic curvature problem (0.1) ( P k (W) P l (W)) 1 k − l = ψ (u , r) on S n must be an origin-centred sphere, where W = (∇ 2 u + u g 0) , ∂ 1 ψ ≥ 0 , ∂ 2 ψ ≥ 0 and at least one of these inequalities is strict. As an application, we establish the uniqueness of solutions to the isotropic Gaussian-Minkowski problem. Finally, we derive the uniqueness result for the following isotropic L p dual Minkowski problem (0.2) u 1 − p r q − n − 1 det ⁡ (W) = 1 on S n , where − n − 1 < p ≤ − 1 and n + 1 ≤ q ≤ n + 1 2 + 1 4 − (1 + p) (n + 1 + p) n (n + 2) . This result utilizes the method developed by Ivaki and Milman [42] and generalizes a result due to Brendle, Choi and Daskalopoulos [10]. [ABSTRACT FROM AUTHOR]

Let (M , g , σ) be an m -dimensional closed spin manifold, with a fixed Riemannian metric g and a fixed spin structure σ ; let S (M) be the spinor bundle over M. The spinorial Yamabe-type problems address the solvability of the following equation D g ψ = f (x) | ψ | g 2 m − 1 ψ , ψ : M → S (M) , x ∈ M where D g is the associated Dirac operator and f : M → R is a given function. The study of such nonlinear equation is motivated by its important applications in Spin Geometry: when m = 2 , a solution corresponds to a conformal isometric immersion of the universal covering M ˜ into R 3 with prescribed mean curvature f ; meanwhile, for general dimensions and f ≡ c o n s t a n t ≠ 0 , a solution provides an upper bound estimate for the Bär-Hijazi-Lott invariant. The aim of this paper is to establish non-compactness results related to the spinorial Yamabe-type problems. Precisely, concrete analysis is made for two specific models on the manifold (S m , g) where the solution set of the spinorial Yamabe-type problem is not compact: 1). the geometric potential f is constant (say f ≡ 1) with the background metric g being a C k perturbation of the canonical round metric g S m , which is not conformally flat somewhere on S m ; 2). f is a perturbation from constant and is of class C 2 , while the background metric g ≡ g S m . [ABSTRACT FROM AUTHOR]

We consider the quotient module Q of the Hardy module H 2 (S) defined by an analytic set M ˜ satisfying certain conditions. Denote d = dim C M ˜. When d = 1 , Q was shown to be 1-essentially normal in [24]. An analogous problem for the case d ≥ 2 was proposed in [24] , which asks whether 2 d -antisymmetric sums of certain module operators are in the trace class. In this paper we solve this problem in the affirmative. In the case d = 1 , we derive a trace formula on Q , which answers another question raised in [24]. [ABSTRACT FROM AUTHOR]

A classical result of Cheng states that the bottom spectrum of complete manifolds of fixed dimension and Ricci curvature lower bound achieves its maximal value on the corresponding hyperbolic space. The paper establishes an analogous result for three-dimensional complete manifolds with scalar curvature lower bound subject to some necessary topological assumptions. The rigidity issue is also addressed and a splitting theorem is obtained for such manifolds with the maximal bottom spectrum. [ABSTRACT FROM AUTHOR]

A fundamental ingredient in the noncommutative geometry program is the notion of KK-duality, often called K-theoretic Poincaré duality, that generalises Spanier-Whitehead duality. In this paper we construct a θ -summable Fredholm module that represents the fundamental class in KK-duality between the stable and unstable Ruelle algebras of a Smale space. To find such a representative, we construct dynamical partitions of unity on the Smale space with highly controlled Lipschitz constants. This requires a generalisation of Bowen's Markov partitions. Along with an aperiodic point-sampling technique we produce a noncommutative analogue of Whitney's embedding theorem, leading to the Fredholm module. [ABSTRACT FROM AUTHOR]

We improve the classical results by Brenner and Thomée on rational approximations of operator semigroups. In the setting of Hilbert spaces, we introduce a finer regularity scale for initial data, provide sharper stability estimates, and obtain optimal approximation rates. Moreover, we strengthen a result due to Egert-Rozendaal on subdiagonal Padé approximations of operator semigroups. Our approach is direct and based on the theory of the B - functional calculus developed recently. On the way, we elaborate a new and simple approach to construction of the B -calculus thus making the paper essentially self-contained. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the following singular stochastic differential equation (SDE) d X t = b (t , X t) d t + d B t , 0 ≤ t ≤ T , X 0 = x ∈ R d , where the drift coefficient b : [ 0 , T ] × R d ⟶ R d is Borel measurable, possibly unbounded and has spatial linear growth. The driving noise B t is a d − dimensional Brownian motion. The main objective of the paper is to establish the existence and uniqueness of a strong solution and a Sobolev differentiable stochastic flow for the above SDE. Malliavin differentiability of the solution is also obtained (cf. [21,23]). Our results constitute significant extensions to those in [31,30,14,21,23] by allowing the drift b to be unbounded. We employ methods from white-noise analysis and the Malliavin calculus. As application, we prove existence of a unique strong Malliavin differentiable solution to the following stochastic delay differential equation d X (t) = b (X (t − r) , X (t , 0 , (v , η)) d t + d B (t) , t ≥ 0 , (X (0) , X 0) = (v , η) ∈ R d × L 2 ([ − r , 0 ] , R d) , with the drift coefficient b : R d × R d → R d is a Borel-measurable function bounded in the first argument and has linear growth in the second argument. [ABSTRACT FROM AUTHOR]

This is the third and the last paper in a series of papers on spectral and scattering theory for the Schrödinger operator on a manifold possessing an escape function, for example a manifold with asymptotically Euclidean and/or hyperbolic ends. Here we discuss the time-dependent scattering theory. A long-range perturbation is allowed, and scattering by obstacles, possibly non-smooth and/or unbounded in a certain way, is included in the theory. We also resolve a conjecture by Hempel–Post–Weder on cross-ends transmissions between two or more ends, formulated in a time-dependent manner. [ABSTRACT FROM AUTHOR]

In this paper we study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. Despite its physical importance, this equation has not received a lot of mathematical attention we think due to the extreme complexity of the relativistic structure of the kernel of the collision operator. In this paper we first largely decompose the structure of the relativistic Landau collision operator. After that we prove the global Entropy dissipation estimate. Then we prove the propagation of any polynomial moment for a weak solution. Lastly we prove the existence of a true weak solution for a large class of initial data. [ABSTRACT FROM AUTHOR]

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 w n = 1 in the classical unilateral shift T , defined as T e n = w n e n + 1 , where { e n } n = 1 ∞ form an orthonormal basis of a complex Hilbert space, by a sequence of i.i.d. random variables { X n } n = 1 ∞ ; that is, w n = X n. 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 FROM AUTHOR]

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 functions E Λ = { e 2 π i λ x : λ ∈ Λ } forms an orthonormal basis/Riesz basis/frame for L 2 (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 L 2 (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 FROM AUTHOR]

Abstract Spatially distributed networks of large size arise in a variety of science and engineering problems, such as wireless sensor networks and smart power grids. Most of their features can be described by properties of their state-space matrices whose entries have indices in the vertex set of a graph. In this paper, we introduce novel algebras of Beurling type that contain matrices on a connected simple graph having polynomial off-diagonal decay, and we show that they are Banach subalgebras of B (ℓ p) , 1 ≤ p ≤ ∞ , the space of all bounded operators on the space ℓ p of all p -summable sequences. The ℓ p -stability of state-space matrices is an essential hypothesis for the robustness of spatially distributed networks. In this paper, we establish the equivalence among ℓ p -stabilities of matrices in Beurling algebras for different exponents 1 ≤ p ≤ ∞ , with quantitative analysis for the lower stability bounds. Admission of norm-control inversion plays a crucial role in some engineering practice. In this paper, we prove that matrices in Beurling subalgebras of B (ℓ 2) have norm-controlled inversion and we find a norm-controlled polynomial with close to optimal degree. Polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks. In this paper, we apply our results on norm-controlled inversion to obtain a polynomial estimate to powers of matrices in Beurling algebras. The polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary Markov chain on a spatially distributed network. [ABSTRACT FROM AUTHOR]

The present paper is a continuation of [30] , it complements the earlier results established in [30] and develops a general framework for establishing averaging principles for more complex random dynamical systems. More precisely, in the present paper, we investigate a multiscale nonautonomous random 2D Navier-Stokes system. In order to study asymptotic behavior of the solutions, we establish Stratonovich-Khasminskii averaging principle for the system. It should be emphasized that the similar scheme also works for the Bogoliubov averaging principle, namely, for the two different kinds of averaging principles, we develop a unified methodological framework to establish them. Then, we apply the Stratonovich-Khasminskii averaging principle to multiscale random nonautonomous 2D Navier-Stokes system perturbed by different random forces, such as strong mixing stationary process, Kac-Stroock approximation process, Donsker approximation process, Wong-Zakai approximation process. These random processes have important applications in physics and other natural sciences. [ABSTRACT FROM AUTHOR]

In this paper we study the isomorphism problem for reduced twisted group and groupoid L p -operator algebras. For a locally compact group G and a continuous 2-cocycle σ we define the reduced σ -twisted L p -operator algebra F λ p (G , σ). We show that if p ∈ (1 , ∞) ∖ { 2 } , then two such algebras are isometrically isomorphic if and only if the groups are topologically isomorphic and the continuous 2-cocyles are cohomologous. For a twist E over an étale groupoid G , we define the reduced twisted groupoid L p -operator algebra F λ p (G ; E). In the main result of this paper, we show that for p ∈ [ 1 , ∞) ∖ { 2 } if the groupoids are topologically principal, Hausdorff, étale and have a compact unit space, then two such algebras are isometrically isomorphic if and only if the groupoids are isomorphic and the twists are properly isomorphic. [ABSTRACT FROM AUTHOR]

In the last few years the authors proved Poincaré and Sobolev type inequalities in Heisenberg groups H n for differential forms in the Rumin's complex. The need to substitute the usual de Rham complex of differential forms for Euclidean spaces with the Rumin's complex is due to the different stratification of the Lie algebra of Heisenberg groups. The crucial feature of Rumin's complex is that d c is a differential operator of order 1 or 2 according to the degree of the form. Roughly speaking, Poincaré and Sobolev type inequalities are quantitative formulations of the well known topological problem whether a closed form is exact. More precisely, for suitable p and q , we mean that every exact differential form ω in L p admits a primitive ϕ in L q such that ‖ ϕ ‖ L q ≤ C ‖ ω ‖ L p . The cases of the norm L p , p ≥ 1 and q < ∞ have been already studied in a series of papers by the authors. In the present paper we deal with the limiting case where q = ∞ : it is remarkable that, unlike in the scalar case, when the degree of the forms ω is at least 2, we can take q = ∞ in the left-hand side of the inequality. The corresponding inequality in the Euclidean setting R N (p = N and q = ∞) was proven by Bourgain & Brezis. [ABSTRACT FROM AUTHOR]

In this paper we study two different weighted isoperimetric inequalities. In the first part of the paper we prove a sharp stability result for the isoperimetric inequality with a log-convex weight. In the second part we analyze the behavior of a negative power weight for the perimeter thus providing a complete picture of the isoperimetric problem in this context. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the number of critical points of solutions of nonlinear elliptic equations. We will deal with the case of non-convex, contractile and non-contractile planar domains. We will prove results on the estimate of their number as well as their index. In some cases we will provide the exact calculation. The toy problem concerns the multi-peak solutions of the Gel'fand problem, namely { − Δ u = λ e u in Ω u = 0 on ∂ Ω , where Ω ⊂ R 2 is a bounded smooth domain and λ > 0 is a small parameter. [ABSTRACT FROM AUTHOR]

In this paper we formulate some conjectures in sub-Riemannian geometry concerning a characterisation of the Koranyi-Kaplan ball in a group of Heisenberg type through the existence of a solution to suitably overdetermined problems. We prove an integral identity that provides a rigidity constraint for one of the two problems. By exploiting some new invariances of these Lie groups, for domains having partial symmetry we solve these problems by converting them to known results for the classical p -Laplacian. [ABSTRACT FROM AUTHOR]

In this paper, we establish an improved decay estimate for the Dirichlet energy of Dir-stationary Q -valued functions. As a direct application of this estimate, we derive a Liouville-type theorem for bounded Dir-stationary Q -valued functions defined on R m. Additionally, in an attempt to establish the continuity of Dir-stationary Q -valued functions, we confirm that such functions exhibit the Lebesgue property at every point within their domain. [ABSTRACT FROM AUTHOR]

A celebrated result by Gidas, Ni & Nirenberg asserts that classical positive solutions to semilinear equations − Δ u = f (u) in a ball vanishing at the boundary must be radial and radially decreasing. In this paper we consider small perturbations of this equation and study the quantitative stability counterpart of this result. [ABSTRACT FROM AUTHOR]

The main results in this paper contribute to bringing to the fore novel underlying connections between the contemporary concepts and methods springing from greedy approximation theory with the well-established techniques of classical Banach spaces. We do that by showing that bounded-oscillation unconditional bases, introduced by Dilworth et al. in 2009 in the setting of their search for extraction principles of subsequences verifying partial forms of unconditionality, are the same as truncation quasi-greedy bases, a new breed of bases that appear naturally in the study of the performance of the thresholding greedy algorithm in Banach spaces. We use this identification to provide examples of bases that exhibit that bounded-oscillation unconditionality is a stronger condition than Elton's near unconditionality. We also take advantage of our arguments to provide examples that allow us to tell apart certain types of bases that verify either debilitated unconditionality conditions or weaker forms of quasi-greediness in the context of abstract approximation theory. [ABSTRACT FROM AUTHOR]

We study the concentration phenomenon of the Lane-Emden equation with vanishing potentials { − Δ u n = W n (x) u n p n , u n > 0 , in Ω , u n = 0 , on ∂ Ω , ∫ Ω p n W n (x) u n p n d x ≤ C , where Ω is a smooth bounded domain in R 2 , W n (x) ≥ 0 are bounded functions with zeros in Ω, and p n → ∞ as n → ∞. A typical example is W n (x) = | x | 2 α with 0 ∈ Ω , i.e. the equation turns to be the well-known Hénon equation. The asymptotic behavior for α = 0 has been well studied in the literature. While for α > 0 , the problem becomes much more complicated since a singular Liouville equation appears as a limit problem. In this paper, we study the case α > 0 and prove a quantization property (suppose 0 is a concentration point) p n | x | 2 α u n (x) p n − 1 + t → 8 π e t 2 ∑ i = 1 k δ a i + 8 π (1 + α) e t 2 c t δ 0 , t = 0 , 1 , 2 , for some k ≥ 0 , a i ∈ Ω ∖ { 0 } and some c ≥ 1. Moreover, for α ∉ N , we show that the blow up must be simple, i.e. c = 1. As applications, we also obtain the complete asymptotic behavior of ground state solutions for the Hénon equation. [ABSTRACT FROM AUTHOR]