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# Multilinear Proofs for Convolution Estimates for Degenerate Plane Curves

Suppose that $\g \in C^2\bigl([0,\infty)\bigr)$ is a real-valued function such that $\g(0)=\g'(0)=0$, and $\g''(t)\approx t^{m-2}$, for some integer $m\geq 2$. Let $\Gamma (t)=\bigl(t,\g(t)\bigr)$, $t>0$, be a curve in the plane, and let $d \lambda =dt$ be a measure on this curve. For a function $f$ on $\bR^2$, let $$Tf(x)=(\lambda *f)(x)=\int_0^{\infty} f\bigl(x-\Gamma(t)\bigr)\,dt, \quad x\in\bR^2 .$$ An elementary proof is given for the optimal $L^p$-$L^q$ mapping properties of $T$.