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

Volume 11, Issue 8, pp 984--993 http://dx.doi.org/10.22436/jnsa.011.08.05 Publication Date: June 09, 2018       Article History

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


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