Research on Fourier integrals of several variables

(1)证明了Calderon-Zygmund型奇异积分和Littlewood-Paley函数的加权弱型（1，1）估计.这些算子由某些粗糙核定义。(2)证明了加权哈代空间上Littlewood-Paley函数的加权估计.另外，还得到了广义Bochner-Riesz算子和广义球面平均的加权估计. (3)对于某些类的伪微分算子，我们证明了L^2_w - L^2_w，L^1_w -L^1_w和H^1_w - L^1_w估计. (4)我们证明了与变旋转曲面相关的某些奇异积分的L^p估计。(5)我们研究了由粗糙核产生的多线性Littlewood-Paley函数。(6)通过对Marcinkiewicz积分的核假设单位球面S^上的LlogL条件，证明了Marcinkiewicz积分的弱（1，1）型估计<n-1>. (7)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。(8)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。(9)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 weighted weak type (1,1) estimates both for the Calderon-Zygmund type singular integrals and for the Littlewood-Paley functions. These operators are defined by certain rough kernels.(2) We proved some weighted estimates for the Littlewood-Paley functions on the weighted Hardy spaces. Also some weighted estimates for the generalized Bochner-Riesz operators and for the generalized spherical means are obtained.(3) For certain classes of pseudo-differential operators, we proved L^2_w - L^2_w, L^1_w - L^<1,∽>_w and H^1_w - L^1_w estimates.(4) We proved the L^p estimates for certain singular integrals associated to the variable surface of revolution.(5) We studied certain multilinear Littlewood-Paley functions arising from rough kernels.(6) We proved the weak type (1,1) estimates for the Marcinkiewicz integrals by assuming for the kernel the LlogL condition on the unit sphere S^<n-1>.(7) 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.(8) 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.(9) 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 surface of revolution, both with H^1 kernels.