Two methods of constructing infinitely many isomorphically distinct Lpspaces have been published. In this article we show that these constructions yield very different spaces and in the process develop methods for dealing with these spaces from the isomorphic viewpoint. We use these methods to give a complete isomorphic classification of the spaces Rp constructed by Bourgain, Rosenthal, and Sch...

We consider classical Tsirelson-type norms of T [An, θ] and their modified versions on lp spaces. We show that for any 1 < p < ∞ there is a constant λp such that considered Tsirelson-type norms do not λp-distort any of subspaces of lp.

For 1 ≤ p ≤ 2, we prove sharp weak-type (p, p) estimates for the BeurlingAhlfors operator acting on the radial function subspaces of Lp(C). A similar sharp Lp result is proved for 1 < p ≤ 2. The results are derived from martingale inequalities which are of independent interest.

For every n ≥ 3, we construct an n-dimensional Banach space which is isometric to a subspace of L1/2 but is not isometric to a subspace of L1. The isomorphic version of this problem (posed by S. Kwapien in 1969) is still open. Another example gives a Banach subspace of L1/4 which does not embed isometrically in L1/2. Note that, from the isomorphic point of view, all the spaces Lq with q < 1 hav...