Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball \B \sub \C^n with its relative logarithmic capacity in \C^n with respect to the same ball \B. An analoguous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of \C^n is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of \psh lemniscates associated to the Lelong class of \psh functions of logarithmic singularities at infinity on \C^n as well as the Cegrell class of \psh functions of bounded Monge-Amp\`ere mass on a hyperconvex domain \W \Sub \C^n. Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of \psh functions.Comment: 25 page

Compound invariants and mixed F-, DF-power spaces

The problems on isomorphic classification and quasiequivalence of bases are studied for the class of mixed $F$-, \DF-power series spaces, {\it i.e.} the spaces of the following kind G(\la,a)=\lim_{p \to \infty} \proj \biggl(\lim_{q \to \infty}\ind \Bigl(\ell_1\bigl(a_i (p,q)\bigr)\Bigr)\biggr), where a_i (p,q)=\exp\bigl((p-\la_i q)a_i\bigr), $p,q \in \N$, and \la =( \la_i)_{i \in \N}, $a=(a_i)_{i \in \N}$ are some sequences of positive numbers. These spaces, up to isomorphisms, are basis subspaces of tensor products of power series spaces of $F$- and \DF-types, respectively. The $m$-rectangle characteristic $\mu_m^{\lambda,a}(\delta,\varepsilon; \tau,t)$, $m \in \N$ of the space G(\la,a) is defined as the number of members of the sequence (\la_i, a_i)_{i \in \N} which are contained in the union of $m$ rectangles $P_k = (\delta_k, \varepsilon_k] \times (\tau_k, t_k]$, $k = 1,2, \ldots, m$. It is shown that each $m$-rectangle characteristic is an invariant on the considered class under some proper definition of an equivalency relation. The main tool are new compound invariants, which combine some version of the classical approximative dimensions (Kolmogorov, Pe{\l}czynski) with appropriate geometrical and interpolational operations under neighborhoods of the origin (taken from a given basis)

Compound invariants and embeddings of Cartesian products

New compound geometric invariants are constructed in order to characterize complemented embeddings of Cartesian products of power series spaces. Bessaga's conjecture is proved for the same class of spaces