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Multilinear Proofs for Convolution Estimates for Degenerate Plane Curves

Published online by Cambridge University Press:  20 November 2018

Jong-Guk Bak
Jong-Guk Bak*
Affiliation:
Department of Mathematics Pohang University of Science and Technology Pohang 790-784 Korea, email: bak@euclid.postech.ac.kr
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Abstract

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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 $T$.

Keywords

42A8542B15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 43 , Issue 1 , 01 March 2000 , pp. 17 - 20
Copyright
Copyright © Canadian Mathematical Society 2000

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