  In this paper, we propose an elementary construction of homogeneous Sobolev
spaces of fractional order on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. This
construction completes the construction of homogeneous Besov spaces on
$\mathcal{S}'_h(\mathbb{R}^n)$ started by Bahouri, Chemin and Danchin on
$\mathbb{R}^n$. We will also extend the treatment done by Danchin and Mucha on
$\mathbb{R}^n_+$, and the construction of homogeneous Sobolev spaces of integer
orders started by Danchin, Hieber, Mucha and Tolksdorf on $\mathbb{R}^n$ and
$\mathbb{R}^n_+$. Properties of real and complex interpolation, duality, and
density are discussed. Trace results are also reviewed. Our approach relies
mostly on interpolation theory and yields simpler proofs of some already known
results in the case of Besov spaces. The lack of completeness on the whole
scale will lead to consideration of intersection spaces with decoupled
estimates to circumvent this issue. As standard and simple applications, we
treat the problems of Dirichlet and Neumann Laplacians in these homogeneous
functions spaces.
