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 ${\ell }_r$ for . For $H_b\left({\ell }_r\right)$, the space of entire functions of bounded type in ${\ell }_r$, we prove that $mon{H}_b\left({\ell }_r\right)$ is exactly the Marcinkiewicz sequence space $m_{{\mathrm{\Psi }}_r}$, where the symbol ${\mathrm{\Psi }}_r$ is given by ${\mathrm{\Psi }}_r\left(n\right):=log{\left(n+1\right)}^{1-1∕r}$ for $n\in {\mathbb{N}}_0$.

For the space of $m$-homogeneous polynomials on ${\ell }_r$, we prove that the set of monomial convergence $mon\mathcal{𝒫}\left({}^m{\ell }_r\right)$ contains the sequence space ${\ell }_q$, where $q={\left(m{r}^{\prime }\right)}^{\prime }$. Moreover, we show that for any , the Lorentz sequence space ${\ell }_{q,s}$ lies in $mon\mathcal{𝒫}\left({}^m{\ell }_r\right)$, provided that $m$ is large enough. We apply our results to make an advance in the description of the set of monomial convergence of $H_{\infty }\left(B_{{\ell }_r}\right)$ (the space of bounded holomorphic functions on the unit ball of ${\ell }_r$). As a byproduct we close the gap on certain estimates related to the mixed unconditionality constant for spaces of polynomials over classical sequence spaces.