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

Abstract

Abstract Let L=−ΔHn+V $L=-\Delta_{\mathbb{H}_{n}}+V$ be a Schrödinger operator on the Heisenberg group Hn $\mathbb{H}_{n}$, where the nonnegative potential V belongs to the reverse Hölder class RHq1 $RH_{q_{1}}$ for some q1≥Q/2 $q_{1} \ge Q/2$, and Q is the homogeneous dimension of Hn $\mathbb{H} _{n}$. Let b belong to a new Campanato space Λνθ(ρ) $\Lambda_{\nu }^{ \theta }(\rho )$, and let IβL $\mathcal{I}_{\beta }^{L}$ be the fractional integral operator associated with L. In this paper, we study the boundedness of the commutators [b,IβL] $[b,\mathcal{I}_{\beta }^{L}]$ with b∈Λνθ(ρ) $b \in \Lambda_{\nu }^{\theta }(\rho )$ on central generalized Morrey spaces LMp,φα,V(Hn) $LM_{p,\varphi }^{\alpha ,V}(\mathbb{H}_{n})$, generalized Morrey spaces Mp,φα,V(Hn) $M_{p,\varphi }^{\alpha ,V}(\mathbb{H}_{n})$, and vanishing generalized Morrey spaces VMp,φα,V(Hn) $VM_{p,\varphi }^{\alpha ,V}(\mathbb{H}_{n})$ associated with Schrödinger operator, respectively. When b belongs to Λνθ(ρ) $\Lambda_{\nu }^{\theta }(\rho )$ with θ>0 $\theta >0$, 0<ν<1 $0<\nu <1$ and (φ1,φ2) $(\varphi_{1},\varphi_{2})$ satisfies some conditions, we show that the commutator operator [b,IβL] $[b,\mathcal{I}_{\beta }^{L}]$ is bounded from LMp,φ1α,V(Hn) $LM_{p,\varphi_{1}}^{\alpha ,V}(\mathbb{H}_{n})$ to LMq,φ2α,V(Hn) $LM_{q,\varphi _{2}}^{\alpha ,V}(\mathbb{H}_{n})$, from Mp,φ1α,V(Hn) $M_{p,\varphi_{1}}^{\alpha ,V}( \mathbb{H}_{n})$ to Mq,φ2α,V(Hn) $M_{q,\varphi_{2}}^{\alpha ,V}(\mathbb{H}_{n})$, and from VMp,φ1α,V(Hn) $VM_{p,\varphi_{1}}^{\alpha ,V}(\mathbb{H}_{n})$ to VMq,φ2α,V(Hn) $VM_{q, \varphi_{2}}^{\alpha ,V}(\mathbb{H}_{n})$, 1/p−1/q=(β+ν)/Q $1/p-1/q=(\beta +\nu )/Q$.

Keywords

• Schrödinger operator
• Heisenberg group
• Central generalized Morrey space
• Campanato space
• Fractional integral
• Commutator