Morrey spaces for Schrödinger operators with certain nonnegative potentials, Littlewood–Paley and Lusin functions on the Heisenberg groups


Let \({\mathcal {L}}=-\varDelta _{{\mathbb {H}}^n}+V\) be a Schrödinger operator on the Heisenberg group \({\mathbb {H}}^n\), where \(\varDelta _{{\mathbb {H}}^n}\) is the sublaplacian on \({\mathbb {H}}^n\) and the nonnegative potential V belongs to the reverse Hölder class \(RH_q\) with \(q\ge Q/2\). Here \(Q=2n+2\) is the homogeneous dimension of \({\mathbb {H}}^n\). In this paper the author first introduces a class of Morrey spaces associated with the Schrödinger operator \({\mathcal {L}}\) on \({\mathbb {H}}^n\). Then by using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the Littlewood–Paley function \({\mathfrak {g}}_{{\mathcal {L}}}\) and the Lusin area integral \({\mathcal {S}}_{{\mathcal {L}}}\)(with respect to the heat semigroup \(\{e^{-s{\mathcal {L}}}\}_{s>0}\)) acting on the Morrey spaces. It can be shown that the same conclusions also hold for the operators \({\mathfrak {g}}_{\sqrt{{\mathcal {L}}}}\) and \({\mathcal {S}}_{\sqrt{{\mathcal {L}}}}\) with respect to the Poisson semigroup \(\{e^{-s\sqrt{{\mathcal {L}}}}\}_{s>0}\).

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The author would like to express his deep gratitude to the referee for his/her careful reading, valuable comments and suggestions which made this article more readable.

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Correspondence to Hua Wang.

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Communicated by Maria Alessandra Ragusa.

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Wang, H. Morrey spaces for Schrödinger operators with certain nonnegative potentials, Littlewood–Paley and Lusin functions on the Heisenberg groups. Banach J. Math. Anal. (2020).

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  • Schrödinger operator
  • Littlewood–Paley function
  • Lusin area integral
  • Heisenberg group
  • Morrey spaces
  • Reverse Hölder class

Mathematics Subject Classification

  • 42B20
  • 35J10
  • 22E25
  • 22E30