Necessary and sufficient conditions governing two-weight norm estimates for multiple Hardy and potential operators are presented. Two-weight inequalities for potentials defined on nonhomogeneous spaces are also discussed. Sketches of the proofs for most of the results are given.
We consider two operator space versions of type and cotype, namely -type, -cotype and type (p,H), cotype (q,H) for a homogeneous Hilbertian operator space H and 1 ≤ p ≤ 2 ≤ q ≤ ∞, generalizing “OH-cotype 2” of G. Pisier. We compute type and cotype of some Hilbertian operator spaces and spaces, and we investigate the relationship between a homogeneous Hilbertian space H and operator spaces with cotype (2,H). As applications we consider operator space versions of generalized little Grothendieck’s...
We present a reflexive Banach space with an unconditional basis which is quasi-minimal and tight by range, i.e. of type (4) in Ferenczi-Rosendal's list related to Gowers' classification program of Banach spaces, but in contrast to the recently constructed space of type (4), our space is also tight with constants, thus essentially extending the list of known examples in Gowers' program. The space is defined on the basis of a boundedly modified mixed Tsirelson space with the use of a special coding...
We prove a geometric characterization of Banach space stability. We show that a Banach space X is stable if and only if the following condition holds. Whenever is an ultrapower of X and B is a ball in , the intersection B ∩ X can be uniformly approximated by finite unions and intersections of balls in X; furthermore, the radius of these balls can be taken arbitrarily close to the radius of B, and the norm of their centers arbitrarily close to the norm of the center of B. The preceding condition...