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 $q=(m{r}^{\prime }{)}^{\prime }$. Moreover, we show that for any $q\le s<\infty$, 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.

40 pages