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An ${L}^{p}-{L}^{q}$ estimate for Radon transforms associated to polynomials. (English) Zbl 0980.42008

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

$S\left(x,y\right)=\sum _{d=0}^{n}\sum _{j+k=d}{a}_{jk}{x}^{j}{y}^{k}·$

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

$Rf\left(t,x\right)={\int }_{-\infty }^{\infty }f\left(t+S\left(x,y\right),y\right)\psi \left(t,x,y\right)dy,$

and

$Tf\left(t,x\right)={\int }_{-\infty }^{\infty }f\left(t+S\left(x,y\right),y\right)dy,$

respectively, where $\psi \in {C}_{c}^{\infty }\left({ℝ}^{3}\right)$ is a cutoff function. Let ${\Delta }$ be the closed convex hull of the points $O=\left(0,0\right)$, $A=\left(2/\left(n+1\right)$, $1/\left(n+1\right)\right)$, ${A}^{\text{'}}=\left(n/\left(n+1\right)$, $\left(n-1\right)/\left(n+1\right)\right)$, and ${O}^{\text{'}}=\left(1,1\right)$. When $S\left(x,y\right)$ 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 ${L}^{p}\left({ℝ}^{2}\right)$ to ${L}^{q}\left({ℝ}^{2}\right)$, if $\left(1/p,1/q\right)$ is in the set ${\Delta }$ minus the half-open segments $\left(O,A\right]$ and $\left[{A}^{\text{'}},{O}^{\text{'}}\right)$. They also proved that for $R$ to be bounded from ${L}^{p}\left({ℝ}^{2}\right)$ to ${L}^{q}\left({ℝ}^{2}\right)$, it is necessary that $\left(1/p,1/q\right)$ is in ${\Delta }$, and in the case $n=2,3$ it is known that $R$ is bounded precisely for $\left(1/p,1/q\right)\in {\Delta }$. 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

${\parallel Tf\parallel }_{{L}^{n+1}}\le {C}_{n}|{a}_{1,n-1}{|}^{-1/\left(n+1\right)}{\parallel f\parallel }_{{L}^{\left(n+1\right)/2}},$

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.

##### MSC:
 42B20 Singular and oscillatory integrals, several variables 42B15 Multipliers, several variables 42B30 ${H}^{p}$-spaces (Fourier analysis) 44A12 Radon transform