导　　师： 赵凯

学科专业： G0101

授予学位： 硕士

作　　者： ;

机构地区： 青岛大学

摘　　要： 振荡奇异积分算子由下式定义：tf(x)=p.v.∫rneip(x,y)k(x-y)f(y)dy,这里p(x，y)为rn×rn上的实多项式，k(x-y)为一标准calderón-zygmund核。首先，在平移不变的情形，上述算子与支于低维流形上的奇异积分有关。它也和与扭积相关的heisenberg群(以及其它幂零群)有联系。第三，在奇异radon变换及其应用到-(a)-neumann问题的研究理论中，它可作为典型算子。 f.ricci和e.m.stein证明了t在lp(rn)(1＜p＜∞)上有界，进一步，当核k的条件适当放松时，他们得到一个相似的结论。 本文是f.ricci和e.m.stein的工作的继续。我们首先给出μ-calderón-zygmund核的定义(第一章)，然后证明具有此种核的振荡奇异积分在lp(rn)(1＜p＜∞)上有界，从而在μ的范围可扩大的意义上改进了已有的结果。以此为基础，我们进一步得到一个加权的结果：t在lωp(rn)，(ω∈ap，1＜p＜∞)上也有界。 这一过程的主要困难在于t的局部可能发散．作为三个主要的工具，我们详细给出了关于多项式的不等式(第二章)，vandercorput型估计(第三章)，和多维情形的杨不等式(第四章)。最后，我们证明了本文的主要结果；定理1(第五章)和定理2(第六章)。 The oscillatory integral operator is defined bywhere P/(x, y/) is a real polynomial on R~n x R~n, and K/(x — y/) is a standard Calderon-Zygmund kernel. First, in the translation invariant case, this operator is partly in connection with singular integrals on the lower-dimensional varieties. Also it is connected with the Heisenberg group in relation to twisted convolution /(and generalization of this to other nilpotent groups/). Third, it can be as the model operator occurring in the theory of the singular Radon transforms and their application to the study of the /(?/)-Neumann problem.F.Ricci and E.M.Stein showed that T is bounded on L~p/(R~n/) /(1 < p < ∞/). Furthermore, they obtained a similar theorem for the above operator when the conditions imposed on the kernel K is loosen in a certain extent.As the continuer of the work of F.Ricci E.M.Stem, we define a μ-Calderou-Zygmund kernel /(Chapter 1/) and show that the oscillatory singular integral operator T with this kind of kernel is bounded on L~p/(R~n/) /(1 < p < ∞/), improving the previously known result in the sense that the scope of μ, can be extendable. On the base of that, we obtain a weighted result: T is also bounded on L/_ω~p/(R~n/), /(ω∈ A/_p, 1 < p < ∞/).In this process, the major obstacle lies in the fact that the local part of T may not be converge. As three important tools, some inequalities for polynomials /(Chapter 2/), some estimates of Van Der Corput type /(Chapter 3/), and the general multi-dimentional case of Young's inequality /(Chapter 4/) are presented in detail. Finally, Theorem 1 and Theorem 2 /(Chapter 5, Chapter 6/) as the two major results of this paper are proved.

关 键 词： 振荡奇异积分 核 有界性 权函数

分 类 号： [O174.2]

领　　域： [理学] [理学]