# Regularity of the Radon transform with respect to the original function

Consider a function $$f: \mathbb{R}^{d} \rightarrow \mathbb{R}$$ (whose properties are to be specified). I note $$\mathbb{S}^{d-1}$$ the hypersphere 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.

$$\newcommand{\IR}{\mathbb{R}}$$It is sufficient to consider $$\theta=e_n$$, the $$n$$-th standard basis vector, because the Sobolev norms are all rotationally invariant.
For a compactly supported, smooth functions $$f$$: \begin{align*} \| \partial^k Rf(\cdot,\theta)\|_{L^1(\IR)} &= \int_\IR \left| \partial^k \int_{\IR^{n-1}} f(t\theta+y) dy \right|dt \\&= \int_\IR \left| \int_{\IR^{n-1}} \partial^{(0,...,0,k)} f(t\theta+y) dy\right|dt \\&\leq \int_\IR \int_{\IR^{n-1}} \left| \partial^{(0,...,0,k)} f(t\theta+y) \right| dy dt \\\implies \|Rf(\cdot,\theta)\|_{W^{k,1}(\IR)} &\leq \|f\|_{W^{k,1}(\IR^n)}\end{align*}
The compactly supported, smooth functions are dense in $$W^{k,1}(\IR^n)$$ so that $$f\mapsto Rf(\cdot,\theta)$$ is a continuous operator $$W^{k,1}(\IR^n)\to W^{k,1}(\IR)$$ with norm $$\leq 1$$.
• This result can be extended to hold for any $p \in [1,\infty)$, right ? May 5 at 7:10
• Also, why do you require $f$ be compactly supported in the calculations ? May 6 at 6:52
• @TitouanVayer Hum, why do you have a dimension-dependent multiplicative constant $C=d^{k+1}$ in the RHS of your upper bound in lemma 7 ? The computations in this post show that one can take $C=1$. May 31 at 11:31