Paraexponentials, Muckenhoupt weights, and resolvents of paraproducts

Abstract:

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