The Bochner and Riesz integral representations for the Radon transform (Q1062259)

The classical Radon transform on compact domains of \({\mathbb{R}}^ n\) defines an operator R on two function spaces of particular interest, which are the spaces C and \(L^ 1\) of continuous and of integrable functions (both spaces can be considered as limits of the scale \(L^ p\), \(1<p<\infty).\) The Bochner and Riesz integral representations of R on C and \(L^ 1\), respectively, are studied. As a sequence, R is weakly compact neither on C nor on \(L^ 1\) (in contrast to compactness of R on \(L^ p\), \(1<p<\infty)\). It follows that R maps weak Cauchy sequences into strong ones on \(L^ 1\), whereas on C does not. Finally, \(R:C\to H^{-s}\) turns out nuclear if \(s>n/2\).