Author: Hua Wang 1
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  • 1 School of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, P. R. China
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Let be a Schrödinger operator on the Heisenberg group n, where Δn is the sublaplacian on n 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 n. Assume that {esL}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)sddsesLf(υ)2dυdssQ/2+11/2,

where

Γα(u):={(υ,s)n×(0,+):u1υ<αs},

and

gλ,L*(f)(u):=0nss+u1υ2λsddsesLf(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 1p<, the following inequality holds true:

SL;2j(f)LpnC2jQ/2+2jQ/psL(f)Lp(n).

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(n). In this article, the author also introduces a class of Morrey spaces associated with the Schrödinger operator L on n. 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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