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An $L^p-L^q$ estimate for Radon transforms associated to polynomials. (English) Zbl 0980.42008
Let $S(x,y)$ be a real polynomial of degree $n\ge 2$ defined by $$S(x,y)=\sum_{d=0}^{n}\sum_{j+k=d} a_{jk}x^jy^k.$$ The Radon transforms $R$, and $T$ of $f$ are defined by $$Rf(t,x)=\int_{-\infty}^{\infty}f(t+S(x,y),y)\psi(t,x,y) dy,$$ and $$Tf(t,x)=\int_{-\infty}^{\infty}f(t+S(x,y),y) dy,$$ respectively, where $\psi\in C_c^\infty(\Bbb R^3)$ is a cutoff function. Let $\Delta$ be the closed convex hull of the points $O=(0,0)$, $A=(2/(n+1)$, $1/(n+1))$, $A'=(n/(n+1)$, $(n-1)/(n+1))$, and $O'=(1,1)$. 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 $L^p(\Bbb R^2)$ to $L^q(\Bbb R^2)$, if $(1/p, 1/q)$ is in the set $\Delta$ minus the half-open segments $(O, A]$ and $[A', O')$. They also proved that for $R$ to be bounded from $L^p(\Bbb R^2)$ to $L^q(\Bbb R^2)$, 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 $$\|Tf\|_{L^{n+1}}\le C_n|a_{1,n-1}|^{-1/(n+1)}\|f\|_{L^{(n+1)/2}},$$ 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:
42B20Singular and oscillatory integrals, several variables
42B15Multipliers, several variables
42B30$H^p$-spaces (Fourier analysis)
44A12Radon transform
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Full Text: DOI
References:
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