Consider a function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$ (whose properties are to be specified). I note $\mathbb{S}^{d-1}$ the hypershperehypersphere and the Radon transform of $f$ defined for $(t,\theta) \in \mathbb{R} \times \mathbb{S}^{d-1}$: \begin{equation} Rf(t,\theta):=\int_{x:\langle x,\theta\rangle=t} f(x)dx=\int_{\mathbb{R}^{d}} f(x)\delta_{t-\langle x,\theta\rangle}(x)dx \end{equation} and I also consider the Sobolev norm $\|.\|_{W^{s}(\mathbb{R}^{d})}$ defined by: \begin{equation} \|f\|_{W^{s}(\mathbb{R}^{d})}:=\sum_{|\alpha|\leq s} \int_{\mathbb{R}^{d}} |\partial^{\alpha}f(x)|dx \end{equation} where $\alpha$ is a multi-index and $\partial^{\alpha}$ the weak-derivative. In the same way on $\mathbb{R}$ I define for $f:\mathbb{R} \rightarrow \mathbb{R}$ the norm $\|f\|_{W^{s}(\mathbb{R})}:=\sum_{k=0}^{s} \int_{\mathbb{R}} |f^{(k)}(t)|dt$ where $f^{(k)}$ stands for $\frac{d^{k}}{dt^{k}}f$.
My question is the following: can we relate the Sobolev norm of $\|Rf(\cdot,\theta)\|_{W^{s}(\mathbb{R})}$ for any $\theta \in \mathbb{S}^{d-1}$ with the norm $\|f\|_{W^{s}(\mathbb{R}^{d})}$ of the original function ?
More precisely under which reasonable conditions on $f$ we have something like for $\theta \in \mathbb{S}^{d-1}$ $\|Rf(\cdot,\theta)\|_{W^{s}(\mathbb{R})}\leq C_\theta \|f\|_{W^{s}(\mathbb{R}^{d})}$ for some constant $C_\theta$ ?
I known that there are connections between the regularity of $f$ and $Rf$ for the Sobolev $2$-norms but I am looking for reference in this case.
