Vector Fourier analysis on compact groups and Assiamoua spaces
Yaogan Mensah
TL;DR
This paper addresses extending Fourier analysis from scalar to vector-valued functions on compact groups by leveraging Assiamoua spaces, a family of function spaces tailored for vector-valued Fourier analysis. The authors define and study Assiamoua spaces $\mathscr{S}_p(\widehat{G},A)$ (and their tensor-product reformulation $\mathfrak{B}_p(\widehat{G},A)$), establish isometric mappings like $\widehat{f}:L^2(G,A)\to \mathscr{S}_2(\widehat{G},A)$, and construct Sobolev spaces $H_\gamma^s(G,A)$ based on these spaces. They prove density, duality, and Plancherel-type results, and show how these spaces underpin Sobolev constructions and their potential generalizations, including vector-valued Sobolev spaces and their embeddings. The work lays a foundation for abstract harmonic analysis with vector measures and paves the way for extensions to locally compact groups and more general vector-valued function spaces.
Abstract
This paper shows how a family of function spaces (coined as Assiamoua spaces) plays a fundamental role in the Fourier analysis of vector-valued functions compact groups. These spaces make it possible to transcribe the classic results of Fourier analysis in the framework of analysis of vector-valued functions and vector measures. The construction of Sobolev spaces of vector-valued functions on compact groups rests heavily on the members of the aforementioned family.