Dyadic structure theorems for multiparameter function spaces

• Ji Li ^[1] ; Jill Pipher ^[2] ; Lesley A. Ward ^[3]
  1. [1] Macquarie University

     Macquarie University

     Australia

     
  2. [2] Brown University

     Brown University

     City of Providence, Estados Unidos

     
  3. [3] University of South Australia

     University of South Australia

     Australia

     

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 31, Nº 3, 2015, págs. 767-797
• Idioma: inglés
• DOI: 10.4171/RMI/853
• Texto completo no disponible (Saber más ...)
• Resumen

  • We prove that the multiparameter (product) space BMO of functions of bounded mean oscillation can be written as the intersection of finitely many dyadic product BMO spaces, with equivalent norms, generalizing the one-parameter result of T. Mei. We establish the analogous dyadic structure theorems for the space VMO of functions of vanishing mean oscillation, for Ap weights, for reverse-Hölder weights and for doubling weights. We survey several definitions of VMO and prove their equivalences, in the continuous, dyadic, one-parameter and product cases. In particular, we introduce the space of dyadic product VMO functions. We show that the weighted product Hardy space Hω1 is the sum of finitely many translates of dyadic weighted H1ω, for each A∞ weight ω, and that the weighted strong maximal function is pointwise comparable to the sum of finitely many dyadic weighted strong maximal functions, for each doubling weight ω. Our results hold in both the compact and non-compact cases.