Weighted estimates for the stability operator of the helicoid on slowly varying domains

We consider the poisson problem $\text{wt}\text{mc}Lu=E$ for the operator $\text{wt}\text{mc}L={\mathrm{\Delta }}_{\text{mb}{R}^2}+2{\cosh }^{-2}(s)$ on domains of the form $\mathrm{\Lambda }:=\{(s,z):\mathrm{\mid }s\mathrm{\mid }\le \mathrm{\ell }(z)\}$, where the width $\mathrm{\ell }(z)$ of the domain $\mathrm{\Lambda }$ varies with $z$. We prove the existence of solutions satisfying weighted estimates when the source term satisfies certain natural orthogonality conditions, and when $e^{-\mathrm{\ell }}$ is small and slowly varying.