A decomposition technique for integrable functions with applications to the divergence problem

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain that can be written as $\Omega=\bigcup_{t} \Omega_t$, where $\{\Omega_t\}_{t\in\Gamma}$ is a countable collection of domains with certain properties. In this work, we develop a technique to decompose a function $f\in L^1(\Omega)$, with vanishing mean value, into the sum of a collection of functions $\{f_t-\tilde{f}_t\}_{t\in\Gamma}$ subordinated to $\{\Omega_t\}_{t\in\Gamma}$ such that $Supp\,(f_t-\tilde{f}_t)\subset\Omega_t$ and $\int f_t-\tilde{f}_t=0$. As an application, we use this decomposition to prove the existence of a solution in weighted Sobolev spaces of the divergence problem $\di\uu=f$ and the well-posedness of the Stokes equations on H\"older-$\alpha$ domains and some other domains with an external cusp arbitrarily narrow. We also consider arbitrary bounded domains. The weights used in each case depend on the type of domain.