Monomial convergence on $\ell_r$

Abstract

For $1 < r \le 2$, we study the set of monomial convergence for spaces of holomorphic functions over $\ell_r$. For $ H_b(\ell_r)$, the space of entire functions of bounded type in $\ell_r$, we prove that $\mbox{mon} H_b(\ell_r)$ is exactly the Marcinkiewicz sequence space $m_{\Psi_r}$ where the symbol $\Psi_r$ is given by $\Psi_r(n) := \log(n + 1)^{1 - \frac{1}{r}}$ for $n \in \mathbb N_0$. For the space of $m$-homogeneous polynomials on $\ell_r$, we prove that the set of monomial convergence $\mbox{mon} \mathcal P (^m \ell_r)$ contains the sequence space $\ell_{q}$ where $q=(mr')'$. Moreover, we show that for any $q\leq s<\infty$, the Lorentz sequence space $\ell_{q,s}$ lies in $\mbox{mon} \mathcal P (^m \ell_r)$, provided that $m$ is large enough. We apply our results to make an advance in the description of the set of monomial convergence of $H_{\infty}(B_{\ell_r})$ (the space of bounded holomorphic on the unit ball of $\ell_r$). As a byproduct we close the gap on certain estimates related with the \emph{mixed} unconditionality constant for spaces of polynomials over classical sequence spaces.