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2 Commentationes Mathematicae

2 2008

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

Grala-Michalak J., Michalak A.

Commentationes Mathematicae

2008

Vol. 48, [Z] 2

129-145

Let J be an infinite set. Let X be a real or complex -order continuous rearrangement invariant quasi-Banach function space over ({0, 1}J ,BJ , J ), the product of J copies of the measure space [formula]. We show that if 0 < p < 2 and X contains a function f with the decreasing rearrangement f* such that [...] for every t ε (0, 1), then it contains an isometric copy of the Lebesgue space Lp(J ). Moreover, if X contains a function f such that [...] for every t ε(0, 1), then it contains an isometric copy of the Lebesgue space L2(J ).

On constructions of isometric copies of Lp(0, 1) spaces (0 < p ≤) by stochastic p-stable processes

Grala-Michalak J., Michalak A.

Commentationes Mathematicae

2008

Vol. 48, [Z] 1

3-12

Let S^p = [formula] be a stochastic process on a probability space (Ω, Σ, P) with independent and time homogeneous increments such that [...] is identically distributed as [formula] where Z_p is a given symmetric p-stable distribution. We show that the closed linear hull of Sp forms an isometric copy of the real Lebesgue space Lp(0, 1) in any quasi-Banach space X consisting of P-a.e. equivalence classes of Σ-measurable real functions on Ω equipped with a rearrangement invariant quasi-norm which contains Sp as a subset. It is possible to construct processes S^p for 0 < p ≤ 2 on [0, 1] with the Lebesgue measure. We show also a complex version of the result.