Paraexponentials, Muckenhoupt weights, and resolvents of paraproducts
Cristina Pereyra and Lesley Ward
We analyze the stability of Muckenhoupt's $\textbf{RH}_p^d$ and $\textbf{A}_p^d$ classes of weights under a nonlinear operation, the $\lambda$-operation. We prove that the dyadic doubling reverse Hölder classes $\textbf{RH}_p^d$ are not preserved under the $\lambda$-operation, but the dyadic doubling $A_p$ classes $\textbf{A}_p^d$ are preserved for $0\lt \lambda \lt 1$. We give an application to the structure of resolvent sets of dyadic paraproduct operators.