PreprintCC BY-SA 4.0arXiv
Oscillatory Integral Operators Along the Paraboloid and Applications
Abstract
We prove decay estimates for oscillatory integral operators along (y1, y2, y12+y22) 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 Lp boundedness of an one-parameter Pierce--Yung operator, and Lp boundedness of a bi-parameter Pierce--Yung operator.
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BibTeX
@misc{zhang2026oscillatory,
title = {{Oscillatory Integral Operators Along the Paraboloid and Applications}},
author = {Jianghao Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14841},
eprint = {2610.14841},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14841},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Oscillatory Integral Operators Along the Paraboloid and Applications
AU - Jianghao Zhang
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14841
UR - https://arxiv.org/abs/2610.14841
AB - 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.
ER -