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.


1 Answer 1


$\newcommand{\IR}{\mathbb{R}}$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 $\leq 1$.

  • 1
    $\begingroup$ Thank you ! I do not really catch why we can consider the standard basis vector, could you elaborate ? (it is not clear what you mean by "Sobolev norms are all rotationally invariant") $\endgroup$ Commented Jul 8, 2021 at 17:12
  • $\begingroup$ This result can be extended to hold for any $p \in [1,\infty)$, right ? $\endgroup$
    – dohmatob
    Commented May 5, 2022 at 7:10
  • $\begingroup$ Also, why do you require $f$ be compactly supported in the calculations ? $\endgroup$
    – dohmatob
    Commented May 6, 2022 at 6:52
  • $\begingroup$ Hi @dohmatob I just saw your comment. I used kind of the same result here arxiv.org/pdf/2112.00423.pdf in Lemma 7. Maybe this will be useful for you $\endgroup$ Commented May 31, 2022 at 9:41
  • $\begingroup$ @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 $C=1$. $\endgroup$
    – dohmatob
    Commented May 31, 2022 at 11:31

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