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.

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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