具μ-Calderón-Zygmund核的振荡积分

【作者】 王磊；

【导师】 赵凯；

【作者基本信息】 青岛大学 ， 基础数学， 2006， 硕士

【摘要】 振荡奇异积分算子由下式定义: 这里P(x,y)为R~n×R~n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(?)-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L~ρ(R~n)(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L~ρ(R~n)(1<ρ<∞)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L_ω~ρ(R~n),(ω ∈A_ρ,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),Van Der Corput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)。更多还原

【Abstract】 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.更多还原