Research on Fourier integrals of several variables

(1)通过对Marcinkiewicz积分的核假设单位球面S^上的L log L条件，证明了Marcinkiewicz积分的弱（1，1）型估计<n-1>. (2)证明了光滑核上多项式相位振荡奇异积分的A_1-加权弱（1，1）估计。(3)证明了多项式相位振荡奇异积分的加权弱（1，1）估计，其粗糙核满足一定的Dini条件，且权的条件与Dini条件相对应. (4)证明了欧几里得空间R^n上的多线性乘子算子与环面T^n上的多线性乘子算子之间的转移定理。并给出了这些结果的一些应用。(5)证明了欧几里得空间R^n上的Littlewood-Paley函数与环面T^n上的Littlewood-Paley函数之间的L^P、弱L^P和H^P-L^P估计的迁移定理。并给出了这些结果的一些应用。(6)我们证明了Littlewood-Paley函数沿着曲线的L^P估计和相关的奇异积分，这两个估计都是由粗糙核引起的。作为应用，我们证明了具有H^1核函数的Marcinkiewicz积分沿着曲线的L^P估计和与旋转曲面相关的奇异积分的L ^P估计。

(1) We proved the weak type (1,1) estimates for the Marcinkiewicz integrals by assuming for the kernel the L log L condition on the unit sphere S^<n-1>.(2) We proved the A_1-weighted weak (1,1) estimates for the oscillatory singular integrals of polynomial phase arising from the smooth kernels.(3) We proved the weighted weak (1,1) estimate for the oscillatory singular integrals of polynomial phase with the rough kernels satisfying certain Dini conditions, assuming for the weights the conditions corresponding to the Dini conditions.(4) We proved transference theorems between the multilinear multiplier operators on the Euclid space R^n and the ones on the torus T^n. Also we obtained some applications of these results.(5) We proved transference theorems for the L^P, the weak L^P and H^P-L^P estimates between the Littlewood-Paley functions on the Euclid space R^n and those on the torus T^n. Also we obtained some applications of these results.(6) We proved the L^P estimates for the Littlewood-Paley functions along curves and the related singular integrals, both arising from the rough kernels. As applications, we proved the L^P estimates for the Marcinkiewicz integrals along curves and the singular integrals associated to the surfaces of revolutions, both with H^1 kernels.