Research on Fourier integrals of several variables

(1)考虑一类非正则伪微分算子T_σ，研究了它们在加权triiebel - lizorkin和Besov空间上的有界性问题。特别地，我们实质上放宽了符号σ由于Bourdaud的正则性条件，使得T在Sobolev空间H^s_p上有界。(2)在假定相关的低维极大算子有界的前提下，证明了与变旋转曲面相关的奇异积分算子的Lp有界性。奇异积分由满足一定大小和消去条件的粗糙核定义。作为应用，我们将Grafakos-Stefanov和Fan-Guo-Pan的结果推广到两个变量的R. Fefferman型奇异积分的情况。(3)研究了一类奇异Radon变换的粗糙核奇异积分。我们证明了在外推论证中有用的奇异积分的某些估计。作为一个应用，我们证明了奇异积分在一定的尖尺寸条件下的Lp有界性。(4)证明了沿旋转曲面的非各向同性奇异积分的某些Lp估计（1<p<∞）。奇异积分由广义齐次粗糙核定义。作为一个应用，我们得到了奇异积分在其核上的尖尺寸条件下的Lp有界性。我们还证明了一个三角积分的一个估计，这对研究非各向同性奇异积分是有用的，并暗示了一个包含由不在仿射超平面上的曲线定义的相函数的三角积分的Stein-Wainger估计。(5)考虑沿有限型流形的奇异积分。根据奇异积分算子核的大小条件，证明了奇异积分算子的一个锐利的Lp估计，这对于外推方法的应用是有用的。

(1) We consider certain non-regular pseudo-differential operators T_σ and study the question of their boundedness on the weighted Triebel-Lizorkin and Besov spaces. In particular, we substantially relax the regularity condition on the symbol σ due to Bourdaud for T to be bounded on the Sobolev spaces H^s_p.(2) We prove the Lp boundedness of the singular integral operators associated with a variable surface of revolution assuming the boundedness of related lower dimensional maximal operators. The singular integrals are defined by rough kernels satisfying certain size and cancellation conditions. As an application, we extend a result of Grafakos-Stefanov and Fan-Guo-Pan to the case of a singular integral of R. Fefferman type of two variables.(3) We study singular integrals with rough kernels, which belong to a class of singular Radon transforms. We prove certain estimates for the singular integrals that are useful in an extrapolation argument. As an application, we prove Lp boundedness of the singular integrals under a certain sharp size condition on their kernels.(4) We prove certain Lp estimates (1<p<∞) for non-isotropic singular integrals along surfaces of revolution. The singular integrals are defined by rough kernels with generalized homogeneity. As an application we obtain Lp boundedness of the singular integrals under a sharp size condition on their kernels. We also prove a certain estimate for a trigonometric integral, which is useful in studying non-isotropic singular integrals and implies the Stein-Wainger estimate for a trigonometric integral involving a phase function defined by a curve that does not lie in an affine hyperplane.(5) We consider a singular integral along a manifold of finite type. We prove a sharp Lp estimate for the singular integral operator according to a size condition of its kernel, which is useful in applying an extrapolation method.