Endpoint $L^p$-$L^q$ estimates for some classes of degenerate Radon transforms in $\Real^2$

We study endpoint $L^p$-$L^q$ estimates for the degenerate Radon transforms in $\mathbb R^2$ given by \begin{equation*} Rf(t,x)=\int_{\mathbb R} f(t+S(x,y),y)\psi(x,y)dy \end{equation*} where $\psi$ is a smooth function supported in a small neighborhood of the origin. Under the assumption $S$ is a smooth function satisfying the left and right finite type conditions ($\partial_x\partial_y^n S(0,0)\neq 0\text{ and } \partial_x^m\partial_y S(0,0)\neq 0$ for some $n$, $m\ge 1$), we obtain complete $L^p$-$L^q$ estimates for $R$.

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Published 1 January 2004