Oscillatory integral operators with homogeneous polynomial phases in several variables
Oscillatory integral operators with homogeneous polynomial phases in several variables
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DOI：
10.1016/j.jfa.2006.11.005
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发表时间：
2006-05
影响因子：
1.7
通讯作者：
A. Greenleaf;M. Pramanik;Wan Tang
中科院分区：
文献类型：
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作者：
A. Greenleaf;M. Pramanik;Wan Tang
我们获得振荡积分算子 Tλ 的 L2 衰减估计，其相位函数是 m 次齐次多项式并满足各种通用性假设。对于任何 m，在 (2+2) 维的情况下获得的衰减率是最佳的，而在更高维度中，当 m 足够大时，结果是尖锐的。大 m 的证明本质上是从代数考虑出发的。对于 (2+2) 维的三次方，证明涉及使用 Phong 和 Stein [D.H. Phong，E.M. Stein，简并傅立叶积分算子和 Radon 变换模型，Ann。数学。 （2）140（1994）703-722]。
We obtain L2decay estimates in λ for oscillatory integral operators Tλwhose phase functions are homogeneous polynomials of degree m and satisfy various genericity assumptions. The decay rates obtained are optimal in the case of (2+2)-dimensions for any m, while in higher dimensions the result is sharp for m sufficiently large. The proof for large m follows from essentially algebraic considerations. For cubics in (2+2)-dimensions, the proof involves decomposing the operator near the conic zero variety of the determinant of the Hessian of the phase function, using an elaboration of the general approach of Phong and Stein [D.H. Phong, E.M. Stein, Models of degenerate Fourier integral operators and Radon transforms, Ann. of Math. (2) 140 (1994) 703–722].