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Let \(L=-\Delta_{\mathbb{H}_{n}}+V\) be a Schrödinger operator on the Heisenberg group \(\mathbb{H}_{n}\), where the nonnegative potential V belongs to the reverse Hölder class \(RH_{q_{1}}\) for some \(q_{1} \ge Q/2\), and Q is the homogeneous dimension of \(\mathbb{H} _{n}\). Let b belong to a new Campanato space \(\Lambda_{\nu }^{ \theta }(\rho )\), and let \(\mathcal{I}_{\beta }^{L}\) be the fractional integral operator associated with L. In this paper, we study the boundedness of the commutators \([b,\mathcal{I}_{\beta }^{L}]\) with \(b \in \Lambda_{\nu }^{\theta }(\rho )\) on central generalized Morrey spaces \(LM_{p,\varphi }^{\alpha ,V}(\mathbb{H}_{n})\), generalized Morrey spaces \(M_{p,\varphi }^{\alpha ,V}(\mathbb{H}_{n})\), and vanishing generalized Morrey spaces \(VM_{p,\varphi }^{\alpha ,V}(\mathbb{H}_{n})\) associated with Schrödinger operator, respectively. When b belongs to \(\Lambda_{\nu }^{\theta }(\rho )\) with \(\theta >0\), \(0<\nu <1\) and \((\varphi_{1},\varphi_{2})\) satisfies some conditions, we show that the commutator operator \([b,\mathcal{I}_{\beta }^{L}]\) is bounded from \(LM_{p,\varphi_{1}}^{\alpha ,V}(\mathbb{H}_{n})\) to \(LM_{q,\varphi _{2}}^{\alpha ,V}(\mathbb{H}_{n})\), from \(M_{p,\varphi_{1}}^{\alpha ,V}( \mathbb{H}_{n})\) to \(M_{q,\varphi_{2}}^{\alpha ,V}(\mathbb{H}_{n})\), and from \(VM_{p,\varphi_{1}}^{\alpha ,V}(\mathbb{H}_{n})\) to \(VM_{q, \varphi_{2}}^{\alpha ,V}(\mathbb{H}_{n})\), \(1/p-1/q=(\beta +\nu )/Q\).

Morrey-type estimates for commutator of fractional integral associated with Schrödinger operators on the Heisenberg group