We determine the smallest Schatten class containing all integral operators with kernels in$L$_{p}(L_{p', q})^{symm}, where 2 <$p$∞ and 1≦$q$≦∞. In particular, we give a negative answer to a problem posed by Arazy, Fisher, Janson and Peetre in [1].
Tien-Cuong DinhUniversité Pierre et Marie Curie – Paris 6, Institut de Mathématiques de JussieuNessim SibonyUniversité Paris-Sud, Mathématique – Bâtiment 425
We introduce a notion of super-potential for positive closed currents of bidegree ($p$,$p$) on projective spaces. This gives a calculus on positive closed currents of arbitrary bidegree. We define in particular the intersection of such currents and the pull-back operator by meromorphic maps. One of the main tools is the introduction of structural discs in the space of positive closed currents which gives a “geometry” on that space. We apply the theory of super-potentials to construct Green currents for rational maps and to study equidistribution problems for holomorphic endomorphisms and for polynomial automorphisms.