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The Complete $(L^p,L^p)$ Mapping Properties of Some Oscillatory Integrals in Several Dimensions

Published:2001-10-01
Printed: Oct 2001
• G. Sampson
• P. Szeptycki
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Abstract

We prove that the operators $\int_{\mathbb{R}_+^2} e^{ix^a \cdot y^b} \varphi (x,y) f(y)\, dy$ map $L^p(\mathbb{R}^2)$ into itself for $p \in J =\bigl[\frac{a_l+b_l}{a_l+(\frac{b_l r}{2})},\frac{a_l+b_l} {a_l(1-\frac{r}{2})}\bigr]$ if $a_l,b_l\ge 1$ and $\varphi(x,y)=|x-y|^{-r}$, $0\le r <2$, the result is sharp. Generalizations to dimensions $d>2$ are indicated.
 MSC Classifications: 42B20 - Singular and oscillatory integrals (Calderon-Zygmund, etc.) 46B70 - Interpolation between normed linear spaces [See also 46M35] 47G10 - Integral operators [See also 45P05]