Monomial convergence on ℓr

Abstract

We develop a novel decomposition of the monomials in order to study the set of monomial convergence for spaces of holomorphic functions over er for 1 < r < 2. For Hb(er), the space of entire functions of bounded type in er, we prove that mon Hb (er) is exactly the Marcinkiewicz sequence space m Ψ, where the symbol Ψr is given by Ψr(n): = log(n + 1)1-1/r for n ϵ ℕ0. For the space of m -homogeneous polynomials on er, we prove that the set of monomial convergence mon P(mer) contains the sequence space eq, where q = (mr 1)1 Moreover, we show that for any q < s < ∞, the Lorentz sequence space eq,s lies in mon P(mer), provided that m is large enough. We apply our results to make an advance in the description of the set of monomial convergence of H∞(Bir) (the space of bounded holomorphic functions on the unit ball of tr). As a byproduct we close the gap on certain estimates related to the mixed unconditionality constant for spaces of polynomials over classical sequence spaces.