Paraexponentials, Muckenhoupt weights, and resolvents of paraproducts

We analyze the stability of Muckenhoupt's $\RHp$ and $\Ap$ classes of weights under a nonlinear operation, the $\lb$-operation. We prove that the dyadic doubling reverse H\"older classes $\RHp$ are not preserved under the $\lb$-operation, but the dyadic doubling $A_p$ classes $\Ap$ are preserved for $0<\lb <1$. We give an application to the structure of resolvent sets of dyadic paraproduct operators.