Abstract
The one-dimensional oscillatory integral operator associated to a real analytic phase S is given byTλf(x)=∫−∞∞eiλS(x,y)χ(x,y)f(y)dy. In their fundamental work, Phong and Stein established sharp L2 estimates for Tλ. The goal of this paper is to extend their results to all endpoints. In particular, we obtain a complete characterization for the mapping properties for Tλ on Lp(R). More precisely, we show that ‖Tλf‖p≲|λ|−α‖f‖p holds for some α>0 if and only if (1αp,1αp′) lies in the reduced Newton polygon of S.
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