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

OpenAlex · İngilizce

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 $VM_{q, \varphi_{2}}^{\alpha ,V}(\mathbb{H}_{n})$ , $1/p-1/q=(\beta +\nu )/Q$ .