Multilinear Proofs for Convolution Estimates for Degenerate Plane Curves

[摘要] Suppose that $g in C^2igl([0,infty)igr)$ is a real-valued functionsuch that $g(0)=g'(0)=0$, and $g''(t)approx t^{m-2}$, for some integer $mgeq 2$. Let $Gamma (t)=igl(t,g(t)igr)$, $t>0$, be a curve in theplane, and let $d lambda =dt$ be a measure on this curve. For afunction $f$on $R^2$, let$$Tf(x)=(lambda *f)(x)=int_0^{infty} figl(x-Gamma(t)igr),dt, quad xinR^2 .$$An elementary proof is given for the optimal $L^p$-$L^q$ mapping properties of $T$.

[学科分类] 数学（综合）

[关键词] resolutions;homology;Lie algebras;associative algebras;non-associative algebras;Jacobi identity;leaf-labeled trees;associahedron