Setup To$\newcommand{\sphere}{\mathbb S^{m-1}}$ any $f\in L^2(\mathbb R^m)$ and (almost) any $k>0$, associate the function $F(k) \in \Sigma := L^2(\sphere)$ given by $$ F(k) (\omega ) = k^{\frac{m-1}2} \hat f(k\omega)$$ (That $F(k)$ actually belongs to $ \Sigma$ is already interesting, since $f$ is merely $L^2$, and restriction theorems for the Fourier transform need $L^p$, $p<2$...but lets take that for granted.) This gives us a map $\phi: L^2(\mathbb R^m) \to L^2([0,\infty) , \Sigma)$ defined by $\phi(f) = F$, i.e. $$ \phi(f)(k)(\omega) = k^{\frac{m-1}2} \hat f(k\omega)$$
Problem statement If we define the spaces $(s\in\mathbb R)$ \begin{align} H_s(\mathbb R^n) := \{ u \in \mathcal S'(\mathbb R^m) : (1+|x|^2)^{s/2} u \in L^2 \}, \\ H^s_0((0,\infty),\Sigma) := \text{closure in }H^s((0,\infty),\Sigma) \text{ of }C^\infty_c((0,\infty),\Sigma), \end{align} Why is it true that for $s\in[0,m/2)$, $\phi$ is a bounded as a map $$ \phi : H_s(\mathbb R^n) \to H^s_0((0,\infty),\Sigma)?$$
Here are some things I know about $\phi$. This map is isometric $L^2(\mathbb R^m) \to L^2((0,\infty),\Sigma)$ by the co-area formula $-\!-$ the factor of $k^{(m-1)/2}$ was chosen specifically to rig this up. On the other hand, given $G\in L^2(\mathbb R^+ , \Sigma)$, we can define $\psi : L^2(\mathbb R^+ , \Sigma) \to L^2(\mathbb R^m)$ defined by
$$ \widehat{\psi(G)}(\xi) = \frac1{|\xi|^{(m-1)/2}} G(|\xi|)\left(\frac{\xi}{|\xi|}\right)$$ Then $$\|\psi(G) \|_{L^2(\mathbb R^m)} = \|\widehat{\psi(G)} \|_{L^2(\mathbb R^m)} = \int_{0}^\infty \|G(r,\bullet)\|_{\Sigma}^2 dr = \|G\|_{L^2([0,\infty);\Sigma)} $$ and $ \phi(\psi(G))(k) = G,$ demonstrating that $\phi$ has an inverse and is therefore a unitary operator.
Its also not hard to check that for any $l\in\mathbb R$, $$ \phi[(1-\Delta)^l f](k) = (1+k^2)^l \phi(f)(k).$$ (The operator $(1-\Delta)^l$ is of course defined as $(1+|\xi|^2)^l$ on the Fourier side.)
With the above identity, I tried to use the assumption by looking at the Fourier transform $\mathcal F((1+|x|^2)^{s/2} u ) = (1-\Delta)^{s/2} \hat u \in L^2$...but this led me to some formula that I can't work with and anyway, I don't know how to get any information on $\phi(u)$ using $\phi(\hat u)$...! Plancharel applied at the level of $\mathbb R^m$ seems to make me walk in circles...
Trying a direct proof also seems to go nowhere. I try to handle the $s$-fractional derivative in $k$ by extending the function to be say even in $k$, (I don't know if odd will be better), then take a 1D Fourier transform. But the moment I do this, I don't know how it should interact with the full $R^m$ Fourier transform of $f$.
Any pointers at all would be greatly appreciated.
I'm working off a paper by Kato and Yajima, but it seems to be from a certain lecture notes of Kuroda which is out of print, libraries near me don't have it, and the still-alive publisher didn't respond to my email :(