Sobolev spaces of vector-valued functions on compact groups
Yaogan Mensah
TL;DR
The paper addresses Sobolev spaces of vector-valued functions on a compact group $G$ and the corresponding embedding theorems. It defines $H_\gamma^s(G,E)$ via weighted Fourier coefficients on the unitary dual $\widehat{G}$ with weights $(1+\gamma(\sigma)^2)^s$ and leverages the isometry $f\mapsto (1+\gamma(\sigma)^2)^{s/2}\widehat{f}$ onto $\mathscr{S}_2(\widehat{G},E)$. The main contributions include continuous embeddings $H_\gamma^t(G,E)\hookrightarrow H_\gamma^s(G,E)$ for $t>s$, $H_\gamma^s(G,E)\hookrightarrow L^2(G,E)$, a sufficient condition $\sum_{\sigma} \frac{d_\sigma^3}{(1+\gamma(\sigma)^2)^s}<\infty$ implying $H_\gamma^s(G,E)\hookrightarrow \mathcal{C}(G,E)$ with a norm bound, and embeddings into $L^{\alpha'}(G,E)$ with $\alpha' = \frac{2t}{t-s}$ when $t>s$ via a vector-valued Hausdorff-Young inequality. These results extend Sobolev embedding theory to vector-valued functions on nonabelian compact groups and provide tools for analysis on groups in abstract harmonic analysis.
Abstract
This paper deals with a class of Sobolev spaces of vector-valued functions on a compact group. Some Sobolev embedding theorems are proved.