**Motivating examples:**

Let $V$ be a

**real vector space**with Haar measure $dv$. The fourier transform induces the following topological isomorphism: $$H^s(V,dv) \cong L^2(V^*,(1+|v^*|^2)^sdv^*)$$ The LHS being the $W^{s,2}$-type Sobolev space and the RHS a weighted $L^2$ space on the dual.Consider $S^1$ a

**circle**with haar measure $d\theta$. Fourier transform Induces a topological isomorphism: $$H^s(S^1,d\theta) \cong l^2(\mathbb{Z},(1+|n|^2)^s dn)$$ Here the LHS is again the sobolev space and on the RHS $dn$ is the counting measure on $\mathbb{Z}$ (This example can be generalized to any finite dimensional torus by suitably modifying both sides).

Notice we get the usual Fourier transform isomorphism by putting $s=0$.

Question:Is there ageneralization of the above to non-abelian Fourier transformson (locally?) compact lie groups? (assuming of course the existence of a Plancherel measure).

**Elaboration:**

Suppose $G$ is a compact (connected) lie group (for simplicity), let $dg$ denote the Haar measure and $d\mu$ the Plancherel measure on $\hat{G}$. Does there exist a function $q: \hat{G} \to \mathbb{R}$ (a weight, hopefully $q$ will be just the eigenvalue of the Caisimir) s.t. the non-abelian Fourier transform induces a topological isomorphism

$$H^s(G,dg) \cong \{ T \in\hat \oplus_{\pi \in \hat{G}} End(V_{\pi}): [ \pi \mapsto ||T_{\pi}||_{HS}] \in L^2(\hat{G},q^sd\mu) \}$$

definingSobolev space (on the line or the circle, let alone on the Lie group)? $\endgroup$ – Yemon Choi Mar 7 '18 at 2:32