Interpolation of Hardy spaces of Dirichlet series

Abstract

We initiate the study of interpolation of the Hardy spaces of Dirichlet series. We prove that there are no variants of the classical interpolation theorems in the setting of Hardy spaces on the countably infinite polytorus T∞ as in the case of the Hardy spaces on the finite dimensional torus. As a byproduct of our results we get a similar conclusion for the Hardy spaces of Dirichlet series by using the famous Bohr lift. Our proofs rely on a general method based on interpolation estimates for finite-dimensional Banach spaces. This method is applied to interpolation of injective tensor products and also of spaces of m-homogeneous polynomials.