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Endpoint estimates for one-dimensional oscillatory integral operators. (English) Zbl 1377.42025

The one dimensional oscillatory integral operator associated to a real analytic phase \(S\) is given by \[ T_{\lambda}f(x) = \int_{-\infty}^{\infty} e^{i\lambda S(x,y)} \chi(x,y)f(y) \, dy \] where \(\chi \in C^{\infty}({\mathbb R}^2)\) is supported in a sufficiently small neighborhood of \((x_0,y_0)\) and \(S\) is locally equivalent to its Taylor expansion: \[ S(x,y)= \sum_{p,q, \geq 0} c_{p,q}(x-x_0)^p (y-y_0)^q. \] In the fundamental paper by D. H. Phong and E. M. Stein [Acta Math. 179, No. 1, 105–152 (1997; Zbl 0896.35147)], \(L^2\) boundedness was investigated. The author proves the following. Let \({N}^{*}(S)\) denote the reduced Newton polygon associated to \(S\) at \((x_0,y_0)\), i.e. the convex hull of the union of all quadrants \([p, \infty) \times [q, \infty)\) with \(c_{p,q} \neq 0\) and \((p,q) \in {\mathbb N}^2\). Let \(p>1\) and \(\alpha >0\). Then \[ \| T_{\lambda} f\|_{L^p} \leq C | \lambda |^{-\alpha} \| f \|_{L^p} \] if and only if \(\left( \frac{1}{p\alpha}, \frac{1}{p'\alpha} \right) \in {N}^{*}(S)\).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Citations:

Zbl 0896.35147
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References:

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