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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 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]

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]

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]