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Models of degenerate Fourier integral operators and Radon transforms. (English) Zbl 0833.43004

Let $S\left(x,y\right)$ be a homogeneous polynomial of degree $n$, and consider the oscillatory integral ${T}_{\lambda }$ defined by

${T}_{\lambda }\left(f\right)\left(x\right)={\int }_{-\infty }^{\infty }{e}^{i\lambda S\left(x,y\right)}{\Psi }\left(x,y\right)f\left(y\right)dy$

where $f$ (resp. ${\Psi }$) denotes a smooth compactly supported function on $ℝ$ (resp. ${ℝ}^{2}$). The operator ${T}_{\lambda }$ is closely related to a class of Radon transforms which we describe as follows: The phase $S\left(x,y\right)$ gives rise to a family $\left(t,x\right)\to {\gamma }_{\left(t,x\right)}$ of curves in the plane via

${\gamma }_{\left(t,x\right)}=\left\{\left(s,y\right);s=t+S\left(x,y\right)\right\}$

with curves ${\gamma }_{\left(t,x\right)}$ and suitable densities $d{\mu }_{\left(t,x\right)}$ defined on them; the associated Radon transform $R$ is given by $R\left(f\right)\left(t,x\right)={\int }_{{\gamma }_{\left(t,x\right)}}fd{\mu }_{\left(t,x\right)}·$ 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 $R$.

##### MSC:
 43A32 Other transforms and operators of Fourier type 35S30 Fourier integral operators