A Hardy–Littlewood maximal operator adapted to the harmonic oscillator

• Julian Bailey ^[1]
  1. [1] Australian National University

     Australian National University

     Australia

     
• Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 59, Nº. 2, 2018, págs. 339-373
• Idioma: inglés
• DOI: 10.33044/revuma.v59n2a07
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• Resumen

  • This paper constructs a Hardy–Littlewood type maximal operator adapted to the Schr¨odinger operator L := −∆ + |x| 2 acting on L2 (Rd). It achieves this through the use of the Gaussian grid ∆γ0, constructed by Maas, van Neerven, and Portal [Ark. Mat. 50 (2012), no. 2, 379–395] with the Ornstein-Uhlenbeck operator in mind. At the scale of this grid, this maximal operator will resemble the classical Hardy–Littlewood operator. At a larger scale, the cubes of the maximal function are decomposed into cubes from ∆γ0 nd weighted appropriately. Through this maximal function, a new class of weights is defined, A+p , with the property that for any w ∈ A+p the heat maximal operator associated with L is bounded from Lp(w) to itself. This class contains any other known class that possesses this property. In particular, it is strictly larger than Ap.