On donne des évaluations précises de la croissance modérée des intégrales de fonctions de classe de Nilsson locale dans ${\mathbf{C}}^2$, exprimées par des caractéristiques topologiques des courbes de ramification des intégrands.

We construct $d$-closed and $d{d}^c$-closed positive currents associated to a holomorphic map $\phi$ via cluster points of normalized weighted truncated image currents. They are constructed using analogues of the Ahlfors length-area inequality in higher dimensions. Such classes of currents are also referred to as Ahlfors currents. We give some applications to equidistribution problems in value distribution theory.

For 1 < p < 2 we obtain sharp lower bounds for the uniform norm of products of homogeneous polynomials on $L_p\left(\mu \right)$, whenever the number of factors is no greater than the dimension of these Banach spaces (a condition readily satisfied in infinite-dimensional settings). The result also holds for the Schatten classes ${}_p$. For p > 2 we present some estimates on the constants involved.