Research on Fourier integrals of several variables

(1)证明了Calderon-Zygmund型奇异积分的加权弱型（1,1）估计。这些操作符由某些粗核定义。我们假设内核满足一定的大小条件和消去条件以及Dini类型条件。这些结果将a . Vargas关于齐次核奇异积分的弱（1,1）估计的结果推广到R^n， n【大于或等于】3。同时，它们也是a . Vargas的结果在R^n， n【大于等于】2上的非齐次核的推广。通过对核的类似条件的假设，也给出了Littlewood-Paley函数的加权弱型估计。(2)对于一类伪微分算子，证明了L^2_ω-L^2_ω、L^1_ω-L^<1、∞> ω和H^1_ω-L^1_ω的估计。我们证明了一个符号满足最小正则性条件的伪微分算子的L^2_ω-L^2_ω估计，其中权ω在Muckenhoupt权类的A_1…More。这改进了K. Yabuta的结果。应用Carbery方法证明了L^1_ω-L^<1、∞>_ω和H^1_ω-L^1_ω的估计。(3)研究了与变旋转曲面相关的奇异积分。我们处理了粗糙核情况，其中奇异积分由球面S^<n-1>上的H^1核函数定义。我们证明了1<p <小于等于2奇异积分的L^p有界性，假设某个低维极大算子在L^s上对所有s bbb1都有界。我们还研究了与旋转曲面相关的分数阶积分的（L^p, L^r）有界性。(4)证明了与某些Bochner-Riesz型傅里叶乘数相关的极大函数的一些加权估计。(5)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_σ在Sobolev空间H^s_p （p【大于等于】2）上有界。少

(1)We proved the weighted weak type (1,1) estimates for the Calderon-Zygmund type singular integrals. These operators are defined by certain rough kernels. We assume that the kernel satisfies a certain size condition and a cancellation condition along with a Dini type condition. These results are a generalization to R^n, n【greater than or equal】3, of the results of A. Vargas on the weak (1,1) estimates for the singular integrals with homogeneous kernels. Also, they are a generalization to the case of inhomogeneous kernels on R^n, n【greater than or equal】2, of the results of A. Vargas. The weighted weak type estimates for the Littlewood-Paley functions are also shown by assuming analogous conditions for the kernels.(2)For certain classes of pseudo-differential operators, we proved L^2_ω-L^2_ω, L^1_ω-L^<1,∞>_ω and H^1_ω-L^1_ω estimates. We proved L^2_ω-L^2_ωestimates for a pseudo-differential operators with a symbol satisfying a minimal regularity condition, where the weight ω is in A_1 … More of Muckenhoupt weight class. This improves a result of K. Yabuta. The L^1_ω-L^<1,∞>_ω and H^1_ω-L^1_ω estimates were proved by applying Carbery's method.(3)We studied the singular integrals associated with the variable surfaces of revolution. We treated the rough kernel case where the singular integral is defined by an H^1 kernel function on the sphere S^<n-1>. We proved the L^p boundedness of the singular integral for 1<p【less than or equal】2 assuming that a certain lower dimensional maximal operator is bounded on L^s for all s>1. We also studied the (L^p, L^r) boundedness for the fractional integrals associated with the surfaces of revolution.(4)We proved some weighted estimates for the maximal functions associated with certain Fourier multipliers of Bochner-Riesz type.(5)We considered certain non-regular pseudo-differential operators T_σ and studied the question of their boundedness on the weighted Triebel-Lizorkin and Besov spaces. In particular, we substantially relaxed the regularity condition on the symbol σ due to Bourdaud for T_σ to be bounded on the Sobolev spaces H^s_p (p【greater than or equal】2). Less