Monomial convergence on ℓr

Galicer, Daniel; Mansilla, Martín; Muro, Santiago; Sevilla-Peris, Pablo

doi:10.2140/apde.2021.14.945

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Monomial convergence on $\ell_r$

Daniel Galicer, Martín Mansilla, Santiago Muro and Pablo Sevilla-Peris

Vol. 14 (2021), No. 3, 945–984

DOI: 10.2140/apde.2021.14.945

Abstract

We develop a novel decomposition of the monomials in order to study the set of monomial convergence for spaces of holomorphic functions over $ℓ_{r}$ for $1 < r \leq 2$ . For $H_{b} (ℓ_{r})$ , the space of entire functions of bounded type in $ℓ_{r}$ , we prove that $mon H_{b} (ℓ_{r})$ is exactly the Marcinkiewicz sequence space $m_{Ψ_{r}}$ , where the symbol $Ψ_{r}$ is given by $Ψ_{r} (n) : = log {(n + 1)}^{1 - 1 ∕ r}$ for $n \in ℕ_{0}$ .

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

Keywords

holomorphic function, homogeneous polynomial, monomial convergence, Banach sequence space

Mathematical Subject Classification 2010

Primary: 32A05, 46E50, 46G20, 46G25

Milestones

Received: 12 July 2019

Accepted: 21 November 2019

Published: 18 May 2021

Authors

Daniel Galicer
Departamento de Matemática Facultad de Cs. Exactas y Naturales Universidad de Buenos Aires and IMAS-CONICET Ciudad Universitaria Buenos Aires Argentina

Martín Mansilla
Departamento de Matemática Facultad de Cs. Exactas y Naturales Universidad de Buenos Aires and IMAS-CONICET Ciudad Universitaria Buenos Aires Argentina

Santiago Muro
Facultad de Cs. Exactas, Ingenieria y Agrimensura Universidad Nacional de Rosario and CIFASIS-CONICET Ocampo y Esmeralda Rosario Argentina

Pablo Sevilla-Peris
Institut Universitari de Matemàtica Pura i Aplicada Universitat Politècnica de València València Spain

Vol. 14, No. 3, 2021