Algebra of convolution type operators with continuous data on Banach function spaces

Volume 119 / 2019

Cláudio A. Fernandes, Alexei Yu. Karlovich, Yuri I. Karlovich Banach Center Publications 119 (2019), 157-171 MSC: Primary 47G10; Secondary 46E30, 42C40. DOI: 10.4064/bc119-8

Abstract

We show that if the Hardy–Littlewood maximal operator is bounded on a reflexive Banach function space $X(\mathbb R)$ and on its associate space $X’(\mathbb R)$, then the space $X(\mathbb R)$ has an unconditional wavelet basis. As a consequence of the existence of a Schauder basis in $X(\mathbb R)$, we prove that the ideal of compact operators ${\cal K}(X(\mathbb R))$ on the space $X(\mathbb R)$ is contained in the Banach algebra generated by all operators of multiplication $aI$ by functions $a\in C(\dot {\mathbb R})$, where $\dot{ \mathbb R}=\mathbb R\cup\{\infty\}$, and by all Fourier convolution operators $W^0(b)$ with symbols $b\in C_X(\dot{ \mathbb R})$, the Fourier multiplier analogue of $C(\dot{ \mathbb R})$.

Authors

  • Cláudio A. FernandesCentro de Matemática e Aplicações
    Departamento de Matemática
    Faculdade de Ciências e Tecnologia
    Universidade Nova de Lisboa
    Quinta da Torre
    2829–516 Caparica, Portugal
    e-mail
  • Alexei Yu. KarlovichCentro de Matemática e Aplicações
    Departamento de Matemática
    Faculdade de Ciências e Tecnologia
    Universidade Nova de Lisboa
    Quinta da Torre
    2829–516 Caparica, Portugal
    e-mail
  • Yuri I. KarlovichCentro de Investigación en Ciencias
    Instituto de Investigación en Ciencias Básicas y Aplicadas
    Universidad Autónoma del Estado de Morelos
    Av. Universidad 1001
    Col. Chamilpa
    C.P. 62209 Cuernavaca
    Morelos, México
    e-mail

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