Models of degenerate Fourier integral operators and Radon transforms (Q1842645)

Let \(S(x,y)\) be a homogeneous polynomial of degree \(n\), and consider the oscillatory integral \(T_\lambda\) defined by \[ T_\lambda(f) (x) = \int^\infty_{-\infty} e^{i\lambda S(x,y)} \Psi(x,y) f(y) dy \] where \(f\) (resp. \(\Psi\)) denotes a smooth compactly supported function on \(\mathbb{R}\) (resp. \(\mathbb{R}^2\)). The operator \(T_\lambda\) is closely related to a class of Radon transforms which we describe as follows: The phase \(S(x,y)\) gives rise to a family \((t,x) \to \gamma_{(t,x)}\) of curves in the plane via \[ \gamma_{(t,x)} = \{(s,y); s = t + S(x,y)\} \] with curves \(\gamma_{(t,x)}\) and suitable densities \(d \mu_{(t,x)}\) defined on them; the associated Radon transform \(R\) is given by \(R(f) (t,x) = \int_{\gamma_{(t,x)}} fd\mu_{(t,x)}.\) In this paper the authors give necessary and sufficient conditions for optimal \(L^2\) estimates of the operator \(T_\lambda\). Next, by viewing \(R\) as a pseudo-differential operator whose symbol is the operator-valued function \(\lambda \to T_\lambda\) they deduce the \(L^p - L^q\) boundedness and Sobolev regularity of the transform \(R\).