An optimal multiplier theorem for Grushin operators in the plane, I

• Gian Maria Dall Ara ^[2] ; Alessio Martini ^[1]
  1. [1] Polytechnic University of Turin

     Polytechnic University of Turin

     Torino, Italia

     
  2. [2] Research Unit Scuola Normale Superiore, Pisa
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 3, 2023, págs. 897-974
• Idioma: inglés
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• Resumen

  • Let L D @ 2 x V .x/@2 y be the Grushin operator on R2 with coefficient V W R ! Œ0; 1/. Under the sole assumptions that V .x/ ' V .x/ ' xV 0 .x/ and x 2 jV 00.x/j . V .x/, we prove a spectral multiplier theorem of Mihlin–Hörmander type for L, whose smoothness requirement is optimal and independent of V . The assumption on the second derivative V 00 can actually be weakened to a Höldertype condition on V 0 . The proof hinges on the spectral analysis of one-dimensional Schrödinger operators, including universal estimates of eigenvalue gaps and matrix coefficients of the potent