[For the entire collection see Zbl 0722.00051.]
This is an overview of the authors’ forthcoming paper. Let \(X=G/K\), a rank one symmetric space of noncompact type an \(\Xi=G/MN\), the space of horocycles in \(X\). Then as Helgason has developed harmonic analysis on \(X\) and \(\Xi\), the horocycle transform and its dual are defined by \[ Rf(kM,a)=\int_ N f(kan)dn\quad (f\in C_ c^ \infty(X)),\qquad R^*h(g)=\int_ K h(gkMN)dk\quad (h\in L^ 1_{loc}(\Xi)). \] Under suitable convolution structures on \(X\) and \(\Xi\), \(R\) and \(R^*\) are related by a filtered backprojection type identity of the form: \(f*R^*\phi=R^*(Rf*\phi)\) (\(f\in L^ 1_ c(X)\) and a \(K\)-invariant \(\phi\in C(\Xi)\)). Then, by calculating the explicit formula of the fundamental solution of the adjoint equation: \(R^*\phi=\Phi\), the authors deduce the inversion formula: \(f=CR^*\Lambda Rf\), where \(\Lambda\) is an integro-differential operator related to the Plancherel measure. The operator is of the form: \(\Lambda=e^{-\rho}\circ\Lambda^ 1\circ e^ \rho\) and \(\Lambda^ 1=(d/ds)P(d/ds)\circ H\) (\(P\) is a polynomial and \(H\) a p.v. convolution operator). Then, taking into account the interaction between the behaviors of \(R^*\) and \(\Lambda\), they find a cancellation property stated as follows: for the characteristic function \(\chi(t)\) of the ball \(B(0,t)\) in \(X\) of radius \(t\) and centered at the origin \[ P(d/ds)\circ H[(d/dt)e^{- \rho(\cdot)}R\chi_ t(\cdot)](s)=0\quad (| s|<t). \] This property plays a key role in their range theorem when they treat the \(K\)- invariant functions on \(X\). Actually, they first obtain a continuous extension of the Abel transform to the space \(D^ s_ K(X)\) (\(s\leq 0\)) consisting of \(L^ 2\) functions \(f\) on \(X\) such that \(\widehat f(\lambda)| c(\lambda)|^{-2s}\in L^ 2(\mathbb{R},| c(\lambda)|^{-2}d\lambda)\), where \(\widehat f\) is the spherical transform of \(f\), and then, give a Paley-Wiener theorem for \(D^ s_ K(X)\) that characterizes the spherical transform \(\widehat f\) of \(f\in D^ s_ K(X)\) with \(\text{supp}(f)\subset\overline{B}(0,R)\).