Estimates for the commutators of Riesz transforms related to Schrödinger-type operators

Let ℒ 2 = ( - Δ ) 2 + V 2 {\mathcal{L}_{2}=(-\Delta)^{2}+V^{2}} be the Schrödinger-type operator on ℝ n {\mathbb{R}^{n}} ( n ≥ 5 {n\geq 5} ), let H ℒ 2 1 ⁢ ( ℝ n ) {H^{1}_{\mathcal{L}_{2}}(\mathbb{R}^{n})} be the Hardy space related to ℒ 2 {\mathcal{L}_{2}} , and let BMO θ ⁢ ( ρ ) {\mathrm{BMO}_{\theta}(\rho)} be the BMO-type space introduced by Bongioanni, Harboure and Salinas. In this paper, we investigate the boundedness of commutator [ b , T α , β , j ] {[b,T_{\alpha,\beta,j}]} , which is generated by the Riesz transform T α , β , j = V 2 ⁢ α ⁢ ∇ j ⁡ ℒ 2 - β {T_{\alpha,\beta,j}=V^{2\alpha}\nabla^{j}\mathcal{L}_{2}^{-\beta}} , j = 1 , 2 , 3 {j=1,2,3} , and b ∈ BMO θ ⁢ ( ρ ) {b\in\mathrm{BMO}_{\theta}(\rho)} . Here, 0 < α ≤ 1 - j 4 {0<\alpha\leq 1-\frac{j}{4}} , j 4 < β ≤ 1 {\frac{j}{4}<\beta\leq 1} , β - α = j 4 {\beta-\alpha=\frac{j}{4}} , and the nonnegative potential V belongs to both the reverse Hölder class RH s {\mathrm{RH}_{s}} with s ≥ n 2 {s\geq\frac{n}{2}} and the Gaussian class associated with ( - Δ ) 2 {(-\Delta)^{2}} . The L p {L^{p}} boundedness of [ b , T α , β , j ] {[b,T_{\alpha,\beta,j}]} is obtained, and it is also shown that [ b , T α , β , j ] {[b,T_{\alpha,\beta,j}]} is bounded from H ℒ 2 1 ⁢ ( ℝ n ) {H^{1}_{\mathcal{L}_{2}}(\mathbb{R}^{n})} to weak L 1 ⁢ ( ℝ n ) {L^{1}(\mathbb{R}^{n})} .