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

Abstract

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 $\{e^{-s\sqrt{{\mathcal {L}}}}\}_{s>0}$.