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。这些算子是由某些粗糙核来定义的。我们假设核满足一定的大小条件和抵消条件以及Dini类型条件。这些结果是A.Vargas关于齐次核奇异积分的弱(1，1)估计结果在R^n，n[大于或等于]3的推广。同时，它们也是A.Vargas的结果在R^n，n[大于或等于]2上的非齐次核的情况的推广。(2)对于某些拟微分算子，我们证明了L^2_ω-L^2_ω，L^1_ω-L^&lt；1，∞&gt；_ω和H^1_ω-L^1_ω估计。证明了带符号的伪微分算子满足极小正则性条件的L^2_ω-L^2_ω估计，其中权ω在A_1…中更多的是Muckenoupt重量级。这改进了K.Yabuta的结果。利用Carberg方法证明了L^1_ω-L^&lt；1，∞&gt；_ω和H^1_ω-L^1_ω估计。(3)研究了与可变旋转曲面有关的奇异积分。我们讨论了球面S^&lt；n-1&gt；上的H^1核函数定义奇异积分的粗核情形。假设某一低维极大算子在L^S上对所有S有界，我们证明了1&lt；p[小于等于]2的奇异积分的S^p有界性。我们还研究了与旋转曲面有关的分数次积分的(L^p，L^r)有界性。(4)证明了与某些Bochner-Riesz型傅立叶乘子相关的极大函数的一些加权估计。(5)考虑了一类非正则伪微分算子T_Besov，研究了它们在加权Triebel-Lizorkin和Besov空间上的有界性问题。特别地，我们大大放宽了由于Bourdaud而导致的符号σ上的正则性条件，即T_σ在Sobolev空间H^S_p(p[大于或等于]2)上有界。较少

(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