Sharp Lp estimates for Schrödinger groups on spaces of homogeneous type

• The Anh Bui ^[1] ; Piero D'Ancona ^[3] ; Fabio Nicola ^[2]
  1. [1] Macquarie University

     Macquarie University

     Australia

     
  2. [2] Polytechnic University of Turin

     Polytechnic University of Turin

     Torino, Italia

     
  3. [3] Università di Roma La Sapienza

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 2, 2020, págs. 455-484
• Idioma: inglés
• DOI: 10.4171/rmi/1136
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• Resumen

  • We prove an Lp estimate ∥e−itLφ(L)f∥p≲(1+|t|)s∥f∥p,t∈R,s=n∣∣12−1p∣∣ for the Schrödinger group generated by a semibounded, self-adjoint operator L on a metric measure space X of homogeneous type (where n is the doubling dimension of X). The assumptions on L are a mild Lp0→Lp′0 smoothing estimate and a mild L2→L2 off-diagonal estimate for the corresponding heat kernel e−tL. The estimate is uniform for φ varying in bounded sets of S(R),or more generally of a suitable weighted Sobolev space.