Multilinear Proofs for Convolution Estimates for Degenerate Plane Curves

Bak, Jong-Guk

Canadian mathematical bulletin, Tome 43 (2000) no. 1, pp. 17-20

Résumé

Suppose that $\gamma \in {{C}^{2}}\left( \left[ 0,\infty\right]) \right)$ is a real-valued function such that $\gamma \left( 0 \right)\,=\,{\gamma }'\left( 0 \right)\,=\,0$ , and ${\gamma }''\left( t \right)\,\approx \,{{t}^{m-2}}$ , for some integer $m\,\ge \,2$ . Let $\Gamma \left( t \right)\,=\,\left( t,\,\gamma \left( t \right) \right),\,t\,>\,0$ , be a curve in the plane, and let $d\text{ }\!\!\lambda\!\!\text{ }\,\text{=}\,dt$ be a measure on this curve. For a function $f$ on ${{\mathbf{R}}^{2}}$ , let $$Tf\left( x \right)\,=\,\left( \text{ }\lambda \text{ }*f \right)\left( x \right)=\int_{0}^{\infty }{f\left( x-\Gamma \left( t \right) \right)dt,\,\,x\in {{\mathbf{R}}^{2}}}.$$ An elementary proof is given for the optimal ${{L}^{p}}-{{L}^{q}}$ mapping properties of $T$ .

Détail

DOI : 10.4153/CMB-2000-002-2

Mots-clés : 42A85, 42B15