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

Eroglu, Ahmet ¹ ; Gadjiev, Tahir ² ; Namazov, Faig ³

¹ Department of Mathematics, Nigde Omer Halisdemir University, Nigde, Turkey
² Institute of Mathematics and Mechanics, NAS of Azerbaijan, AZ1141 Baku, Azerbaijan
³ Baku State University, AZ1141 Baku, Azerbaijan

Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 8, p. 984-993.

Résumé

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 $M_{q,\varphi_2}^{\alpha,V}$, $1/p-1/q=\beta/Q$.

Détail

DOI : 10.22436/jnsa.011.08.05

Classification : 22E30, 35J10, 42B35, 47H50
Keywords: Schrödinger operator, Heisenberg group, central generalized Morrey space, fractional integral, commutator, BMO

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