Sobolev sheaves on the plane

In this paper, we show that for any integer $k\in \mathbb{N}$ there exists a Sobolev sheaf (in the sense of Lebeau) on any definable site of ${\mathbb{R}}^2$ that agrees with Sobolev spaces on cuspidal domains. We also provide a complete computation of the cohomology of these sheaves using the notion of ‘Good direction’ introduced by Valette. This paper serves as an introduction to a more general project on the sheafification of Sobolev spaces in higher dimensions.