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In this paper, we present some unexpected answers to two problems with longstanding interest. More specifically, we show that a bounded primitive function of an asymptotically almost periodic function from R to a Banach space X is remotely almost periodic if and only if c 0 ⊄ X. This gives a natural analogy of the classical Bohl-Bohr-Amerio-Kadets theorem for asymptotically almost periodic functions. Based on this result, we succeed in extending the classical Loomis theorem in [13] to asymptotically almost periodic case without any ergodic conditions, and apply our Loomis type theorems to abstract Cauchy problems. Moreover, we give an example of inhomogeneous Schrödinger equation with asymptotically almost periodic coefficient, whose solution is not asymptotically almost periodic but remotely almost periodic. This demonstrates that remotely almost periodic functions is the natural class for solutions to some partial differential equations, which is similar to a remarkable phenomenon revealed by Shen and Yi (1998) [20] using dynamical system approach. [ABSTRACT FROM AUTHOR]

Let L be a non-negative self-adjoint operator acting on L 2 (X) where X is a space of homogeneous type with a dimension n. Suppose that the heat kernel of L satisfies the Davies-Gaffney estimates of order m ≥ 2. Let H L 1 (X) be the Hardy space associated with L. In this paper we obtain the sharp endpoint estimate for the Schrödinger group e i t L associated with L such that ‖ (I + L) − n / 2 e i t L f ‖ L 1 (X) + ‖ (I + L) − n / 2 e i t L f ‖ H L 1 (X) ≤ C (1 + | t |) n / 2 ‖ f ‖ H L 1 (X) , t ∈ R for some constant C = C (n , m) > 0 independent of t. We further apply our result to provide the sharp estimate for Schrödinger group of the Kohn Laplacian □ b on polynomial model domains treated by Nagel–Stein [41] , where e − t □ b satisfies only the second order Davies-Gaffney estimates. Moreover, when the heat kernel of L satisfies a Gaussian upper bound, by a duality and interpolation argument, it gives a new proof of a recent result of [13] for sharp endpoint L p -Sobolev bound for e i t L : ‖ (I + L) − s e i t L f ‖ L p (X) ≤ C (1 + | t |) s ‖ f ‖ L p (X) , t ∈ R , s ≥ n | 1 2 − 1 p | for every 1 < p < ∞ , which extends the classical results due to Miyachi ([39,40]) for the Laplacian on the Euclidean space R n. [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]

In this paper, we investigate the complexity of random dynamical systems. Assume that ϕ is a continuous random dynamical system on Polish space X over an ergodic Polish system (Ω , F , P , θ). Let K be a ϕ -invariant random set with compact ω -section K (ω) such that ϕ (ω) (K (ω)) = K (θ ω). We proved that if h top (ϕ , K) > 0 , then there exists chaotic phenomenon (strong second type of distributional chaos) in this random dynamical system. [ABSTRACT FROM AUTHOR]

Let X be a smooth Artin stack with properly stable good moduli space X → π X. The purpose of this paper is to prove that a simple geometric criterion can often characterize when the moduli space X is smooth and the morphism π is flat. [ABSTRACT FROM AUTHOR]

Boundary effects are crucial for dynamics of dilute charged gases governed by the Vlasov-Poisson-Boltzmann (VPB) system. In this paper, we study the existence and regularity of solutions to the VPB system with soft potential in a bounded convex domain with in-flow boundary condition. We establish the existence of strong solutions in the time interval [ 0 , T ] for an arbitrary given T > 0 when the initial distribution function is near an absolute Maxwellian. Our contribution is based on a new weighted energy estimate in some W 1 , p space and L x 3 L v 1 + space for soft potential. By using the classical L 2 – L ∞ method and bootstrap argument, we extend the local solutions from small time scale to large time scale. [ABSTRACT FROM AUTHOR]

Combinatorial vector fields on simplicial complexes introduced by Robin Forman constitute a combinatorial analogue of classical flows. They have found numerous and varied applications in recent years. Yet, their formal relationship to classical dynamical systems has been less clear. In this paper we prove that for every combinatorial vector field on a finite simplicial complex X one can construct a semiflow on the underlying polytope X which exhibits the same dynamics. The equivalence of the dynamical behavior is established in the sense of Conley-Morse graphs and uses a tiling of the topological space X which makes it possible to directly construct isolating blocks for all involved isolated invariant sets based purely on the combinatorial information. [ABSTRACT FROM AUTHOR]

In this paper we establish inclusions and noninclusions between various Hardy type spaces on noncompact Riemannian manifolds M with Ricci curvature bounded from below, positive injectivity radius and spectral gap. Our first main result states that, if L is the positive Laplace–Beltrami operator on M , then the Riesz–Hardy space H R 1 (M) is the isomorphic image of the Goldberg type space h 1 (M) via the map L 1 / 2 (I + L) − 1 / 2 , a fact that is false in R n. Specifically, H R 1 (M) agrees with the Hardy type space X 1 / 2 (M) recently introduced by the first three authors; as a consequence, we prove that H R 1 (M) does not admit an atomic characterisation. Noninclusions are mostly proved in the special case where the manifold is a Damek–Ricci space S. Our second main result states that H R 1 (S) , the heat Hardy space H H 1 (S) and the Poisson–Hardy space H P 1 (S) are mutually distinct spaces, a fact which is in sharp contrast to the Euclidean case, where these three spaces agree. [ABSTRACT FROM AUTHOR]

Consider a connected reductive algebraic group G and a symmetric subgroup K. Let X = K / B K × G / P be a double flag variety of finite type, where B K is a Borel subgroup of K , and P a parabolic subgroup of G. A general argument shows that the orbit space C X / K inherits a natural action of the Hecke algebra H = H (K , B K) of double cosets via convolutions. However, it is a quite different problem to find out the explicit structure of the Hecke module. In this paper, for the double flag variety of type AIII, we determine the explicit action of H on C X / K in a combinatorial way using graphs. As a by-product, we also get the description of the representation of the Weyl group on C X / K as a direct sum of induced representations. [ABSTRACT FROM AUTHOR]

Our goal in this paper is to advance the state of the art of the topic of uniqueness of unconditional basis. To that end we establish general conditions on a pair (X , Y) formed by a quasi-Banach space X and a Banach space Y which guarantee that every unconditional basis of their direct sum X ⊕ Y splits into unconditional bases of each summand. As application of our methods we obtain that, among others, the spaces H p (T d) ⊕ T (2) and H p (T d) ⊕ ℓ 2 , for p ∈ (0 , 1) and d ∈ N , have a unique unconditional basis (up to equivalence and permutation). [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to study the problem concerning isomorphic embeddings of ideals of compact operators due to Arazy (see [5, Problem C] and [6, Problem (B)]), which is closely related to Pełczyński's problem of uniqueness of symmetric structure in such ideals. In particular, we show that for a wide class of symmetric sequence spaces E , if C F ↪ S E , then F = ℓ 2 and E has a subspace isomorphic to ℓ 2 , where C X is the symmetric ideal of compact operators on the Hilbert space ℓ 2 induced by a symmetric sequence space X and S X = (⊕ n = 1 ∞ M n) C X . This extends results due to Arazy [5,6]. As a consequence, we contribute to the Arazy–Pełczyński problem on the uniqueness of symmetric structures [5,6]. In particular, we can deal with the case when E has a subspace isomorphic to ℓ 2 or c 0 , which was left untreated in [5,6]. [ABSTRACT FROM AUTHOR]

In the present paper we introduce and study the Lipschitz retractional structure of metric spaces. This topic was motivated by the analogous projectional structure of Banach spaces, a topic that has been thoroughly investigated. The more general metric setting fits well with the currently active theory of Lipschitz free spaces and spaces of Lipschitz functions. Among our applications we show that the Lipschitz free space F (X) is a Plichko space whenever X is a Plichko Banach space. Our main results include two examples of metric spaces. The first one M contains two points { 0 , 1 } such that no separable subset of M containing these points is a Lipschitz retract of M. The second example fails the analogous property for arbitrary infinite density. Finally, we introduce the metric version of the concept of locally complemented Banach subspace, and prove some metric analogues to the linear theory. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study slit Sierpiński spaces and slit Menger curves. These metric spaces appear as limits of slit domains in R n , n ≥ 2 , equipped with the inner metric. Using these slit spaces we construct the first examples of metric spaces homeomorphic to the universal Menger curve and higher dimensional Sierpiński spaces, which are quasisymmetrically (QS) co-Hopfian. Here a metric space X is QS co-Hopfian if every quasisymmetric embedding of X into itself is onto. It is also shown that the collection of quasisymmetric equivalence classes of slit Menger curves is uncountable. These results answer a problem and generalize results of Merenkov from [34]. [ABSTRACT FROM AUTHOR]