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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 countablecollection of domains with certain properties. In this work, we develop atechnique to decompose a function $f\in L^1(\Omega)$, with vanishing meanvalue, 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 $\intf_t-\tilde{f}_t=0$. As an application, we use this decomposition to prove theexistence 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. Wealso consider arbitrary bounded domains. The weights used in each case dependon the type of domain.

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