Application de l'étude de certaines formes linéaires aléatoires au plongement d'espaces de Banach dans des espaces $L^p$

Random embedding of $${\ell_p^n}$$ into $${\ell_r^N}$$

“…Let us fix p ∈ (0, 2), and recall that for any fixed r , such that 0 < r < p, the operator T was defined in (4). The main theorem that we shall prove is the following:…”

“…Also, the result of Naor and Zvavitch has been extended by Bernués and López-Valdes [1] who proved that n p C(log n,η,r ) → (1+η)n r when r ≤ 1. Even if it is not the main object of that paper, there is an important related subject concerning embedding subspaces of L p into L r , which started with the work of [5,4,26,20]. Of course, this was extended to the finite dimensional setting and we refer the reader to [3,32,33], where the embedding of general finite dimensional subspaces of L p into N r were studied, and to the survey [14].…”

Embeddings of non-commutative L p -spaces into non-commutative L 1 -spaces, 1 < p < 2

“…The theory of embeddings of classical L p -spaces started with the work of Bretagnolle, Dacuhna-Castelle and Krivine [BrDK] and later Dacuhna-Castelle [BrD] between 1966 and1969. In particular, they found embeddings of L q -spaces and also Orlicz spaces in L p -spaces based on the Lévy-Khintchine representation of infinite divisible random variables.…”

Reflexive subspaces of some Orlicz spaces

“…Bretagnolle and Dacunha-Castelle showed in [3] that an Orlicz space L φ embeds into L 1 (meaning that there exists an isomorphism of this space onto a subspace of L 1 ) if and only if φ is 2-concave (recall that a function f is r-concave if f (x 1/r ) is concave). If φ is an Orlicz function whose conjugate φ * satisfies the condition ∆ 0 (see below for the definition), then φ is equivalent, for every r > 1, to an r-concave Orlicz function (Proposition 4) and hence L φ embeds into L 1 .…”