On constructions of isometric embeddings of nonseparable L p spaces, 0 < p 2

Jolanta Grala-Michalak, Artur Michalak
2008 unpublished
Let J be an infinite set. Let X be a real or complex σ-order continuous rearrangement invariant quasi-Banach function space over ({0, 1} J , B J , λ J), the product of J copies of the measure space ({0, 1}, 2 {0,1} , 1 2 δ 0 + 1 2 δ 1). We show that if 0 < p < 2 and X contains a function f with the decreasing rearrangement f * such that f * (t) t − 1 p for every t ∈ (0, 1), then it contains an isometric copy of the Lebesgue space L p (λ J). Moreover, if X contains a function f such that f * (t)
more » ... f such that f * (t) | ln(t)| for every t ∈ (0, 1), then it contains an isometric copy of the Lebesgue space L 2 (λ J). 2000 Mathematics Subject Classification: Primary 46E30; Secondary 46B09, 46B25, 60J30. In [11] Kadec showed that any sequence of independent symmetric p-stable distributions on a probability space (Ω, Σ, P) span an isometric copy of the sequence space l p in the real Lebesgue space L r (P) for every 1 r < p < 2. This result remains valid for every 0 < r < p < 2 (see [27]). For p = 2 the subspace is iso-metric to l 2 in L q (P) for every 0 < q < ∞. Applying stochastic processes based on p-stable distributions Bretagnolle, Dacunha-Castelle and Krivine in [3] showed that if 1 r p 2, then L p (ν) is isometric to a subspace of L r (ν) where ν is the Lebesgue measure on [0, 1]. They also showed that if 1 r < p 2, then every L p (µ)-space is isomorphic to a subspace of an L r (P)-space for some probability measure P. The reader may find a self-contained probabilistic proof of the fact that L p (ν) is isometric to a subspace of L r (ν) for every 0 < r < p 2 in [13]. In [17] Lindenstrauss and Pee lczy´nskilczy´nski showed, applying the techniques of L p-spaces and the Kadec result, that if 1 r p 2, then every L p-space is isomorphic to a subspace of an L r (µ)-space for some measure µ. One may check, applying the properties of ultraproducts of L p-spaces for 0 < p < 1 (see [23] and [4]), that the above result remains valid for every 0 < r p 2 (see also [9, p. 55] and [6]). Johnson, Maurey,
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