English

Abstract

Let − (n + 1) < m ⩽ − (n + 1) (1 − ϱ) and let $${T_a} \in {\cal L}_{ϱ}^m,\delta $$ be pseudo-differential operators with symbols a(x, ξ) ∈ ℝn × ℝn, where 0 < ϱ ⩽ 1,0 ⩽ δ < 1 and δ ⩽ ϱ. Let μ, λ be weights in Muckenhoupt classes Ap, ν = (μλ−1)1/p for some 1 < p < ∞. We establish a two-weight inequality for commutators generated by pseudo-differential operators Ta with weighted BMO functions b ∈ BMOν, namely, the commutator [b, Ta] is bounded from Lp(μ) into Lp(λ). Furthermore, the range of m can be extended to the whole m ⩽ − (n + 1)(1 − ϱ).