Cohomologie feuillet\'ee du flot affine de Reeb sur la vari\'et\'e de Hopf ${\mathbb{S}}^n\times {\mathbb{S}}^1$

We determine explicitly the foliated cohomology $H_{\mathcal{F}}^{\ast }(M)$ of the affine Reeb flow $\mathcal{F}$ on the Hopf manifold ${\mathbb{S}}^n\times {\mathbb{S}}^1$. The vector space $H_{\mathcal{F}}^1(M)$ contains exactly the obstructions to solve the cohomological equation $X\cdot f=g$ where $f$ and $g$ are $C^{\mathrm{\infty }}$-functions and $X$ is any non singular vector field defining the foliation $\mathcal{F}$. The topological dual of $H_{\mathcal{F}}^1(M)$ is the space of distributions invariant by $X$.