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### Hyperplane conjecture for quotient spaces of Lp.

Forum mathematicum

### Comparing gaussian and Rademacher cotype for operators on the space of continuous functions

Studia Mathematica

We prove an abstract comparison principle which translates gaussian cotype into Rademacher cotype conditions and vice versa. More precisely, let 2 < q < ∞ and T: C(K) → F a continuous linear operator. (1) T is of gaussian cotype q if and only if ${\left({\sum }_{k}{\left(\left(\parallel T{x}_{k}{\parallel }_{F}\right)/\left(\surd log\left(k+1\right)\right)\right)}^{q}\right)}^{1/q}\le c\parallel {\sum }_{k}{\varepsilon }_{k}{x}_{k}{\parallel }_{{L}_{2}\left(C\left(K\right)\right)}$, for all sequences ${\left({x}_{k}\right)}_{k\in ℕ}\subset C\left(K\right)$ with ${\left(\parallel T{x}_{k}\parallel \right)}_{k=1}^{n}$ decreasing. (2) T is of Rademacher cotype q if and only if $\left({\sum }_{k}{\left(\parallel T{x}_{k}{\parallel }_{F}\surd \left({\left(log\left(k+1\right)\right)}^{q}\right)\right)}^{1/q}\le c\parallel {\sum }_{k}{g}_{k}{x}_{k}{\parallel }_{{L}_{2}\left(C\left(K\right)\right)}$, for all sequences ${\left({x}_{k}\right)}_{k\in ℕ}\subset C\left(K\right)$ with ${\left(\parallel T{x}_{k}\parallel \right)}_{k=1}^{n}$ decreasing. Our method allows a restriction to a fixed number of vectors and complements the corresponding results of Talagrand.

### Characterization of weak type by the entropy distribution of r-nuclear operators

Studia Mathematica

The dual of a Banach space X is of weak type p if and only if the entropy numbers of an r-nuclear operator with values in a Banach space of weak type q belong to the Lorentz sequence space ${\ell }_{s,r}$ with 1/s + 1/p + 1/q = 1 + 1/r (0 < r < 1, 1 ≤ p, q ≤ 2). It is enough to test this for Y = X*. This extends results of Carl, König and Kühn.

### Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities

Studia Mathematica

We determine the set of all triples 1 ≤ p,q,r ≤ ∞ for which the so-called Marcinkiewicz-Zygmund inequality is satisfied: There exists a constant c≥ 0 such that for each bounded linear operator $T:{L}_{q}\left(\mu \right)\to {L}_{p}\left(\nu \right)$, each n ∈ ℕ and functions ${f}_{1},...,{f}_{n}\in {L}_{q}\left(\mu \right)$, $\left(ʃ\left({\sum }_{k=1}^{n}|T{f}_{k}{|}^{r}{\right)}^{p/r}{d\nu \right)}^{1/p}\le c\parallel T\parallel \left(ʃ\left({\sum }_{k=1}^{n}|{f}_{k}{|}^{r}{{\right)}^{q/r}d\mu \right)}^{1/q}$. This type of inequality includes as special cases well-known inequalities of Paley, Marcinkiewicz, Zygmund, Grothendieck, and Kwapień. If such a Marcinkiewicz-Zygmund inequality holds for a given triple (p,q,r), then we calculate the best constant c ≥ 0 (with the only exception:...

### Volume ratios in ${L}_{p}$-spaces

Studia Mathematica

There exists an absolute constant ${c}_{0}$ such that for any n-dimensional Banach space E there exists a k-dimensional subspace F ⊂ E with k≤ n/2 such that $in{f}_{ellipsoid\epsilon \subset {B}_{E}}{\left(vol\left({B}_{E}\right)/vol\left(\epsilon \right)\right)}^{1/n}\le {c}_{0}in{f}_{zonoidZ\subset {B}_{F}}{\left(vol\left({B}_{F}\right)/vol\left(Z\right)\right)}^{1/k}$ . The concept of volume ratio with respect to ${\ell }_{p}$-spaces is used to prove the following distance estimate for $2\le q\le p<\infty$: $su{p}_{F\subset {\ell }_{p},dimF=n}in{f}_{G\subset {L}_{q},dimG=n}d\left(F,G\right){\sim }_{{c}_{pq}}{n}^{\left(q/2\right)\left(1/q-1/p\right)}$.

### On weak (r,2)-summing operators and weak Hilbert spaces

Studia Mathematica

### On some matrix Inequalities in Banach Spaces

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales

### Integral mappings and the principle of local reflexivity for noncommutative ${L}^{1}$-spaces.

Annals of Mathematics. Second Series

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