# Fractional integral associated to Schrödinger operator on the Heisenberg groups in central generalized Morrey spaces

Volume 11, Issue 8, pp 984--993 Publication Date: June 09, 2018       Article History
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### Authors

Ahmet Eroglu - Nigde Omer Halisdemir University, Department of Mathematics, Nigde, Turkey
Tahir Gadjiev - Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ1141 Baku, Azerbaijan
Faig Namazov - Baku State University, AZ1141 Baku, Azerbaijan

### Abstract

Let $L=-\Delta_{\mathbb{H}_n}+V$ be a Schrödinger operator on the Heisenberg groups $\mathbb{H}_n$, where the non-negative potential $V$ belongs to the reverse Hölder class $RH_{Q/2}$ and $Q$ is the homogeneous dimension of $\mathbb{H}_n$. Let $b$ belong to a new $BMO_{\theta}(\mathbb{H}_n,\rho)$ space, and let ${\cal I}_{\beta}^{L}$ be the fractional integral operator associated with $L$. In this paper, we study the boundedness of the operator ${\cal I}_{\beta}^{L}$ and its commutators $[b,{\cal I}_{\beta}^{L}]$ with $b \in BMO_{\theta}(\mathbb{H}_n,\rho)$ on central generalized Morrey spaces $LM_{p,\varphi}^{\alpha,V}(\mathbb{H}_n)$ and generalized Morrey spaces $M_{p,\varphi}^{\alpha,V}(\mathbb{H}_n)$ associated with Schrödinger operator. We find the sufficient conditions on the pair $(\varphi_1,\varphi_2)$ which ensures the boundedness of the operator ${\cal I}_{\beta}^{L}$ from $LM_{p,\varphi_1}^{\alpha,V}(\mathbb{H}_n)$ to $LM_{q,\varphi_2}^{\alpha,V}(\mathbb{H}_n)$ and from $M_{p,\varphi_1}^{\alpha,V}(\mathbb{H}_n)$ to $M_{q,\varphi_2}^{\alpha,V}(\mathbb{H}_n)$, $1/p-1/q=\beta/Q$. When $b$ belongs to $BMO_{\theta}(\mathbb{H}_n,\rho)$ and $(\varphi_1,\varphi_2)$ satisfies some conditions, we also show that the commutator operator $[b,{\cal I}_{\beta}^{L}]$ is bounded from $LM_{p,\varphi_1}^{\alpha,V}(\mathbb{H}_n)$ to $LM_{q,\varphi_2}^{\alpha,V}(\mathbb{H}_n)$ and from $M_{p,\varphi_1}^{\alpha,V}$ to $M_{q,\varphi_2}^{\alpha,V}$, $1/p-1/q=\beta/Q$.

### Keywords

• Schrödinger operator
• Heisenberg group
• central generalized Morrey space
• fractional integral
• commutator
• BMO

•  22E30
•  35J10
•  42B35
•  47H50

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