On homogeneous Sobolev and Besov spaces on the whole and the half space

Anatole Gaudin (1)

1 I2M - Institut de Mathématiques de Marseille

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 extends the construction of homogeneous Besov spaces on $\EuScript{S} ′_h (\mathbb{R}^n)$ 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 completness on the whole scale will lead to consideration of intersection spaces with decoupled estimates to circumvent this issue. As standard applications, we treat the problems of Dirichlet and Neumann Laplacians in these homogeneous functions spaces.

Mots clés: Homogeneous Sobolev spaces; Homogeneous Besov spaces; Homogeneous tempered distributions; Traces; Interpolation of non-complete spaces; Half-space; Dirichlet and Neumann Laplacians

Domaines: Analyse fonctionnelle [math.FA]; Equations aux dérivées partielles [math.AP]

Anatole Gaudin. On homogeneous Sobolev and Besov spaces on the whole and the half space. 2022. ⟨hal-03850461v2⟩