Paraexponentials, Muckenhoupt weights, and resolvents of paraproducts

Abstract

We analyze the stability of Muckenhoupt’s R H p d \mathbf {RH}_{\mathbf {p}}^{\mathbf {d}} and A P d \mathbf {A}_{\mathbf {P}}^{\mathbf {d}} classes of weights under a nonlinear operation, the λ \lambda -operation. We prove that the dyadic doubling reverse Hölder classes R H p d \mathbf {RH}_{\mathbf {p}}^{\mathbf {d}} are not preserved under the λ \lambda -operation, but the dyadic doubling A p A_p classes A P d \mathbf {A}_{\mathbf {P}}^{\mathbf {d}} are preserved for 0 ≤ λ ≤ 1 0\leq \lambda \leq 1 . We give an application to the structure of resolvent sets of dyadic paraproduct operators.