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

摘要

Let (L=-Delta_{mathbb{H}_{n}}+V) be a Schr??dinger operator on the Heisenberg group (mathbb{H}_{n}), where the nonnegative potential V belongs to the reverse H??lder 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_{u }^{ heta }(ho )), and let (mathcal{I}_{eta }^{L}) be the fractional integral operator associated with L. In this paper, we study the boundedness of the commutators ([b,mathcal{I}_{eta }^{L}]) with (b in Lambda_{u }^{heta }(ho )) on central generalized Morrey spaces (LM_{p,arphi }^{lpha ,V}(mathbb{H}_{n})), generalized Morrey spaces (M_{p,arphi }^{lpha ,V}(mathbb{H}_{n})), and vanishing generalized Morrey spaces (VM_{p,arphi }^{lpha ,V}(mathbb{H}_{n})) associated with Schr??dinger operator, respectively. When b belongs to (Lambda_{u }^{heta }(ho )) with (heta 0), (0u 1) and ((arphi_{1},arphi_{2})) satisfies some conditions, we show that the commutator operator ([b,mathcal{I}_{eta }^{L}]) is bounded from (LM_{p,arphi_{1}}^{lpha ,V}(mathbb{H}_{n})) to (LM_{q,arphi _{2}}^{lpha ,V}(mathbb{H}_{n})), from (M_{p,arphi_{1}}^{lpha ,V}( mathbb{H}_{n})) to (M_{q,arphi_{2}}^{lpha ,V}(mathbb{H}_{n})), and from (VM_{p,arphi_{1}}^{lpha ,V}(mathbb{H}_{n})) to (VM_{q, arphi_{2}}^{lpha ,V}(mathbb{H}_{n})), (1/p-1/q=(eta +u )/Q).