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

Anatole Gaudin

Vol. 6 (2024), No. 2, 343–404
Abstract

We propose an elementary construction of homogeneous Sobolev spaces of fractional order on n and +n in the scope of the treatment of nonlinear partial differential equations. This construction extends the construction of homogeneous Besov spaces on 𝒮h(n) started by Bahouri, Chemin and Danchin on n. We will also extend the treatment done by Danchin and Mucha on +n, and the construction of homogeneous Sobolev spaces of integer orders started by Danchin, Hieber, Mucha and Tolksdorf on n and +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 for our function spaces with high regularities will lead to the consideration of the intersection with a complete space to enforce the behavior of low frequencies. From this point one performs decoupled estimates, in order to obtain results similar to the one obtained in the case of low regularities.

As standard and simple applications, we treat the problems of Dirichlet and Neumann Laplacians in these homogeneous function spaces.

Keywords
homogeneous Sobolev spaces, homogeneous Besov spaces, homogeneous tempered distributions, traces, interpolation of noncomplete spaces, half-space, Dirichlet and Neumann Laplacians
Mathematical Subject Classification
Primary: 42B37, 46B70, 46E35
Secondary: 35J05
Milestones
Received: 2 September 2023
Revised: 7 December 2023
Accepted: 22 December 2023
Published: 29 June 2024
Authors
Anatole Gaudin
Aix-Marseille Université, CNRS, I2M, Marseille
France