Calderón reproducing formulas and applications to Hardy spaces

• Pascal Auscher ^[3] ; Alan G.R. McIntosh ^[1] ; Andrew J. Morris ^[2]
  1. [1] Australian National University

     Australian National University

     Australia

     
  2. [2] University of Oxford

     University of Oxford

     Oxford District, Reino Unido

     
  3. [3] Université de Paris-Sud

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 31, Nº 3, 2015, págs. 865-900
• Idioma: inglés
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• Resumen

  • We establish new Calderón holomorphic functional calculus whilst the synthesising function interacts with D through functional calculus based on the Fourier transform. We apply these to prove the embedding HpD(∧T∗M)⊆Lp(∧T∗M), 1≤p≤2, for the Hardy spaces of differential forms introduced by Auscher, McIntosh and Russ, where D=d+d∗ is the Hodge–Dirac operator on a complete Riemannian manifold M that has doubling volume growth. This fills a gap in that work. The new reproducing formulas also allow us to obtain an atomic characterisation of H1D(∧T∗M). The embedding HpL⊆Lp, 1≤p≤2, where L is either a divergence form elliptic operator on Rn, or a nonnegative self-adjoint operator that satisfies Davies–Gaffney estimates on a doubling metric measure space, is also established in the case when the semigroup generated by the adjoint −L∗ is ultracontractive.