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arXiv · 2610.14841

Oscillatory Integral Operators Along the Paraboloid and Applications

Jianghao Zhang
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Abstract

We prove decay estimates for oscillatory integral operators along $(y_1, y_2, y_1^2+y_2^2)$ in the finite field and the continuous settings. In the continuous setting, our result allows a bi-parameter maximal modulation, which extends the one-parameter version previously established by Pierce and Yung. Our approach is different from theirs. Then we discuss three applications of our estimates: a power saving bound for certain pattern finding problem in finite fields, a short approach to $L^p$ boundedness of an one-parameter Pierce--Yung operator, and $L^p$ boundedness of a bi-parameter Pierce--Yung operator.

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Jianghao Zhang. 2026-10-10. Oscillatory Integral Operators Along the Paraboloid and Applications. https://arxiv.org/abs/2610.14841

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