Let $\mathcal{L}$=space of all lines $L$. The X-ray transform is defined here: https://en.wikipedia.org/wiki/X-ray_transform $Xf:\mathcal{L}\to \mathbb{R}$ is defined by: $Xf(L)=\int_{t \in L}f(t)|dt|$.

In this definition, the X-ray transform $Xf$ of a function $f$ is defined only for compactly supported continuous functions $f$ on $\mathbb{R}^d$.

Note that $X:\mathcal{C}_c(\mathbb{R}^d)\to \mathcal{S}$, where $\mathcal{S}$ is the space of continuous scalar valued functions on the set $\mathcal{L}$ of all lines L.

My questions are: 1) Under these conditions, is the transform operator $X$ bounded?

2) If we instead assume that $f$ is a rapidly decaying function, i.e., both $f, \nabla f \to 0$ at infinity, what can we say about the boundedness of $X$?

I'd very much appreciate a detailed answer! Thank you!



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