3
$\begingroup$

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.

$\endgroup$
6
$\begingroup$

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

$\endgroup$
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") $\endgroup$ – Titouan Vayer Jul 8 at 17:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.