Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 57: Issue 4
Morrey Spaces Related to Schrödinger Operators with Certain Nonnegative Potentials and Littlewood–Paley–Stein Functions on the Heisenberg groups
Author: Hua Wang 1
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  • 1 School of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, P. R. China
DOI:
https://doi.org/10.1556/012.2020.57.4.1477
Publication Date:
17 Dec 2020
Online Publication Date:
17 Dec 2020
Article Category:
Research Article
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Let be a Schrödinger operator on the Heisenberg group ℍn, where Δℍn is the sublaplacian on ℍn and the nonnegative potential V belongs to the reverse Hölder class Bq with q∈[Q/2,+∞). Here  Q=2n+2 is the homogeneous dimension of ℍn. Assume that {e−sL}s>0 is the heat semigroup generated byL. The Lusin area integral SL;α and the Littlewood–Paley–Stein function gλ,L* associated with the Schrödinger operator L are defined, respectively, by

SL;α(f)(u):=∬Γα(u)sddse−sLf(υ)2dυdssQ/2+11/2,

where

Γα(u):={(υ,s)∈ℍn×(0,+∞):u−1υ<αs},

and

gλ,L*(f)(u):=∫0∞∫ℍnss+u−1υ2λsddse−sLf(υ)2dυdssQ/2+11/2 ,

Where λ∈(0,+∞) is a parameter. In this article, the author shows that there is a relationship between SL;α and the operator gλ,L* and for any 1≤p<∞, the following inequality holds true:

SL;2j(f)Lpℍn≤C2jQ/2+2jQ/psL(f)Lp(ℍn).

Based on this inequality and known results for the Lusin area integral SL;1, the author establishes the strong-type and weak-type estimates for the Littlewood–Paley–Stein function gλ,L* on Lp(ℍn). In this article, the author also introduces a class of Morrey spaces associated with the Schrödinger operator L on ℍn. By using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the operator gλ,L* acting on the Morrey spaces for an appropriate choice of λ>0. It can be shown that the same conclusions hold for the operator gλ,L* on generalized Morrey spaces as well.

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Cover Studia Scientiarum Mathematicarum Hungarica
Studia Scientiarum Mathematicarum Hungarica
Print ISSN:
0081-6906
Online ISSN:
1588-2896
Keywords:
Schrödinger operator; Littlewood–Paley–Stein function; Heisenberg group; Morrey space; reverse Hölder class

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