Research on singular integral operators and function spaces

1)得到了奇异积分点积有限和的多线性算子的H^p估计。所考虑的一类奇异积分包括Calderon-Zygmund奇异积分和任意阶的分数积分。我们也得到了这些多线性算子在其他Hardy型空间中的估计。2)利用尖锐极大函数和洛伦兹空间，定义了一个函数空间，该函数空间由函数的点积构成代数。这个函数空间可以看作是具有临界指标的Sobolev空间的一个变体。3)利用纯实变量方法，证明了在R^n的任意开子集上，可以定义关于加倍测度的加权Hardy空间。这个结果推广了Stromberg和Torchinsky在b施普林格讲座笔记#1381,1989.4中给出的结果。我们证明了A.Seeger的广义triiebel - lizorkin空间中函数的可拓性的两个结果。我们得到了Hardy关于傅里叶变换经典定理的推广。

1) We obtained H^p estimate for the multilinear operators which are finite sums of pointwise products of singular integrals. The class of singular integrals considered include the Calderon-Zygmund singular integrals and fractional integrations of arbitrary orders. We also obtained the estimates of those multilinear operators in other Hardy type spaces.2) Using a sharp maximal function and the Lorentz spaces, we defined a function space which forms an algebra with respect to pointwise product of functions. This function space can be considered as a variant of the Sobolev space with the critical index.3) Using purely real variable method, we showed that one can define the weighted Hardy spaces with respect to doubling measures on arbitrary open subset of R^n. This result extends those of Stromberg and Torchinsky which were given in the Springer Lecture Note #1381, 1989.4) We proved two results conserning the extension of functions in A.Seeger's generalized Triebel-Lizorkin spaces.5) We obtained a generalization of Hardy's clasical theorem on the Fourier transform.