*(English)*Zbl 0980.42008

Let $S(x,y)$ be a real polynomial of degree $n\ge 2$ defined by

The Radon transforms $R$, and $T$ of $f$ are defined by

and

respectively, where $\psi \in {C}_{c}^{\infty}\left({\mathbb{R}}^{3}\right)$ is a cutoff function. Let ${\Delta}$ be the closed convex hull of the points $O=(0,0)$, $A=(2/(n+1)$, $1/(n+1))$, ${A}^{\text{'}}=(n/(n+1)$, $(n-1)/(n+1))$, and ${O}^{\text{'}}=(1,1)$. When $S(x,y)$ is a homogeneous polynomial, and ${a}_{1,n-1}\ne 0$ and ${a}_{n-1,1}\ne 0$, Phong and Stein proved that $R$ is bounded from ${L}^{p}\left({\mathbb{R}}^{2}\right)$ to ${L}^{q}\left({\mathbb{R}}^{2}\right)$, if $(1/p,1/q)$ is in the set ${\Delta}$ minus the half-open segments $(O,A]$ and $[{A}^{\text{'}},{O}^{\text{'}})$. They also proved that for $R$ to be bounded from ${L}^{p}\left({\mathbb{R}}^{2}\right)$ to ${L}^{q}\left({\mathbb{R}}^{2}\right)$, it is necessary that $(1/p,1/q)$ is in ${\Delta}$, and in the case $n=2,3$ it is known that $R$ is bounded precisely for $(1/p,1/q)\in {\Delta}$. The author gives a positive result for endpoint estimates in the case $n\ge 4$, and more. His main result is (for not necessarily homogeneous polynomials): if ${a}_{1,n-1}\ne 0$, then there is ${C}_{n}>0$ such that

where ${C}_{n}$ is independent of the coefficients ${a}_{jk}$. Using this, he gets endpoint estimates, not treated by Phong and Stein, in the homogeneous polynomial case.