The classical Banach space $L_1(L_p)$ consists of measurable scalar functions f on the unit square for which

$$ \begin{align*}\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx < \infty.\end{align*} $$

We show that $L_1(L_p) (1 < p < \infty )$ is primary, meaning that whenever $L_1(L_p) = E\oplus F$ , where E and F are closed subspaces of $L_1(L_p)$ , then either E or F is isomorphic to $L_1(L_p)$ . More generally, we show that $L_1(X)$ is primary for a large class of rearrangement-invariant Banach function spaces.

We prove a Khintchine type inequality under the assumption that the sumof Rademacher randomvariables equals zero. We also showa newtail-bound for a hypergeometric random variable.

We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm $\left| X \right|$ of an isotropic log-concave random vector $X\,\in \,{{\mathbb{R}}^{n}},$ stating that for every $t\,\ge \,1$,

$$\mathbb{P}\left( \left| X \right|\,\ge \,ct\sqrt{n} \right)\,\le \,\exp (-t\sqrt{n}).$$

More precisely we show that for any log-concave random vector $X$ and any $p\,\ge \,1$,

$${{(\mathbb{E}{{\left| X \right|}^{p}})}^{1/p}}\,\sim \,\mathbb{E}\left| X \right|\,+\,\underset{z\in {{S}^{n-1}}}{\mathop{\sup }}\,\,{{(\mathbb{E}{{\left| \left\langle z,\,X \right\rangle \right|}^{p}})}^{1/p}}.$$

We prove a self-adjoint analogue of the Marcus–Pisier inequality, comparing the expected value of convex functionals on random reflection matrices and on elements of the Gaussian orthogonal (or unitary) ensemble.

We study the dimension of “random” Euclidean sections of direct sums of normed spaces. We compare the obtained results with results from $[\text{LMS}]$, to show that for the direct sums the standard randomness with respect to the Haar measure on Grassmanian coincides with a much “weaker” randomness of “diagonal” subspaces (Corollary 1.4 and explanation after). We also add some relative information on “phase transition”.

It is proved that a “typical” $n$-dimensional quotient ${{X}_{n}}$ of $l_{1}^{m}$ with $n={{m}^{\sigma }},0<\sigma <1$, has the property

$$\text{Average}\int_{G}{||Tx|{{|}_{{{X}_{n}}}}d{{h}_{G}}(T)\ge \frac{c}{\sqrt{n{{\log }^{3}}n}}\left( n-\int_{G}{|trT|d{{h}_{G}}(T)} \right)},$$

for every compact group $G$ of operators acting on ${{X}_{n}}$, where ${{d}_{G}}(T)$ stands for the normalized Haar measure on $G$ and the average is taken over all extreme points of the unit ball of ${{X}_{n}}$. Several consequences of this estimate are presented.

Let $\mathbb{T}$ denote the unit circle in the complex plane, and let $X$ be a Banach space that satisfies Burkholder’s $\text{UMD}$ condition. Fix a natural number, $N\,\in \,\mathbb{N}$ . Let $\mathcal{P}$ denote the reverse lexicographical order on ${{\mathbb{Z}}^{N}}$ . For each $f\,\in \,{{L}^{1}}({{\mathbb{T}}^{N}},X)$ , there exists a strongly measurable function $\tilde{f}$ such that formally, for all $\mathbf{n}\,\in \,{{\mathbb{Z}}^{N}},\,\hat{\tilde{f}}\,\left( \mathbf{n} \right)\,=\,-i\,{{sgn }_{\mathcal{P}}}\left( \mathbf{n} \right)\hat{f}\left( \mathbf{n} \right)$ . In this paper, we present a summation method for this conjugate function directly analogous to the martingale methods developed by Asmar and Montgomery-Smith for scalar-valued functions. Using a stochastic integral representation and an application of Garling’s characterization of $\text{UMD}$ spaces, we prove that the associated maximal operator satisfies a weak-type (1, 1) inequality with a constant independent of the dimension $N$.

We extend the recent results of R. Latała and O. Guédon about equivalence of ${{L}_{q}}$ -norms of logconcave random variables (Kahane-Khinchin’s inequality) to the quasi-convex case. We construct examples of quasi-convex bodies ${{K}_{n\,}}\subset {{\mathbb{R}}^{n}}$ which demonstrate that this equivalence fails for uniformly distributed vector on ${{K}_{n}}$ (recall that the uniformly distributed vector on a convex body is logconcave). Our examples also show the lack of the exponential decay of the “tail” volume (for convex bodies such decay was proved by M. Gromov and V. Milman).

Several characterizations of weak cotype 2 and weak Hilbert spaces are given in terms of basis constants and other structural invariants of Banach spaces. For finite-dimensional spaces, characterizations depending on subspaces of fixed proportional dimension are proved.