### Cotype for p-Banach spaces

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Under some assumptions on the pair $({A}_{0},{B}_{0})$, we study equivalence between interpolation properties of linear operators and monotonicity conditions for a pair (Y,Z) of rearrangement invariant quasi-Banach spaces when the extreme spaces of the interpolation are ${L}^{\infty}$. Weak and restricted weak intermediate spaces fall within our context. Applications to classical Lorentz and Lorentz-Orlicz spaces are given.

We study the asymptotic behaviour, as n → ∞, of the Lebesgue measure of the set $x\in K:|{P}_{E}\left(x\right)|\le t$ for a random k-dimensional subspace E ⊂ ℝⁿ and an isotropic convex body K ⊂ ℝⁿ. For k growing slowly to infinity, we prove it to be close to the suitably normalised Gaussian measure in ${\mathbb{R}}^{k}$ of a t-dilate of the Euclidean unit ball. Some of the results hold for a wider class of probabilities on ℝⁿ.

We prove sharp end forms of Holmstedt's reiteration theorem which are closely connected with a general form of Gehring's Lemma. Reverse type conditions for the Hardy-Littlewood-Pólya order are considered and the maximal elements are shown to satisfy generalized Gehring conditions. The methods we use are elementary and based on variants of reverse Hardy inequalities for monotone functions.

When dealing with interpolation spaces by real methods one is lead to compute (or at least to estimate) the K-functional associated to the couple of interpolation spaces. This concept was first introduced by J. Peetre (see [8], [9]) and some efforts have been done to find explicit expressions of it for the case of Lebesgue spaces. It is well known that for the couple consisting of L^{1} and L^{∞} on [0, ∞) K is given by K (t; f, L^{1}, L^{∞...
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