Commutators of BMO functions and degenerate Schrödinger operators with certain nonnegative potentials

摘要

Let ${mathcal{L}f(x)=-frac{1}{omega}sum_{i,j} partial_i(a_{i,j}(cdot)partial_jf)(x)+V(x)f(x)}$ with the non-negative potential V belonging to reverse Hölder class with respect to the measure ω(x)dx, where ω(x) satisfies the A 2 condition of Muckenhoupt and a i,j (x) is a real symmetric matrix satisfying ${lambda^{-1}omega(x)|xi|^2le sum^n_{i,j=1}a_{i,j}(x)xi_ixi_jlelambdaomega(x)|xi|^2. }$ We obtain some estimates for ${V^{alpha}mathcal{L}^{-alpha}}$ on the weighted L p spaces and we study the weighted L p boundedness of the commutator ${[b, V^{alpha} mathcal{L}^{-alpha}]}$ when ${bin BMO_omega}$ and 0 α ≤ 1.