DOI: https://doi.org/10.15446/recolma.v50n1.62205

Operator-valued Fourier multipliers on toroidal Besov spaces

Multiplicadores de Fourier operador-valuados sobre espacios de Besov toroidales

Bienvenido Barraza Martínez, Iván González Martínez, Jairo Hernández Monzón¹

¹ Universidad del Norte, Barranquilla, Colombia. bbarraza@uninorte.edu.co, idgonzalez@uninorte.edu.co, jahernan@uninorte.edu.co

Abstract

We prove in this paper that a sequence M: Zⁿ → L(E) of bounded variation is a Fourier multiplier on the Besov space B^s_{p, q}(Tⁿ, E) for s ∈ R, 1 < p < ∞, 1 ≤ q ≤ 1 and E a Banach space, if and only if E is a UMD-space. This extends the Theorem 4.2 in [3] to the n-dimensional case. As illustration of the applicability of this results we study the solvability of two abstract Cauchy problems with periodic boundary conditions.

Keywords: Fourier multipliers, operator-valued symbols, UMD-spaces, toroidal Besov spaces.

2010 Mathematics Subject Classification: 42A45, 47A56.

Resumen

En el presente artículo se prueba que una sucesión M: Zⁿ → L(E) de variación acotada, es un multiplicador de Fourier sobre el espacio de Besov B^s_{p, q}(Tⁿ, E) para s ∈ R, 1 < p < ∞, 1 ≤ q ≤ 1 y E un espacio de Banach, si y solo si, E es un espacio UMD. Este resultado extiende el Teorema 4.2 en [3] al caso n-dimensional. Como ilustración de la aplicabilidad de este resultado, se estudia la solubilidad de dos problemas de Cauchy abstractos con condiciones de frontera periódicas.

Palabras claves: Multiplicadores de Fourier, símbolos operador-valuados, espacios UMD, espacios de Besov toroidales.

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(Recibido en septiembre de 2015. Aceptado en mayo de 2016)