| 摘要: | 這篇文章的主要目的是討論變量核奇異積分算子的加權有界性。在文章的開頭我們簡單的介紹這個理論的發展，並點出我們研究這個主題的動機。在第二章中，我們介紹 $A_p$ 權及加權哈弟空間的定義，並整理出一些性質，這些性質在我們證明的過程中是不可或缺的。 我們將變量核的條件分成兩類。第一類是考慮在核的兩個變量上都給平滑條件，而第二類是考慮將核的平滑條件都集中在第二個變量上。第三章就是討論奇異積分算子在第一類核條件下的 $L^p_w$ 及 $H^p_w-L^p_w$ 有界性。 第二類核的條件在第二個變量上給了很高的平滑性，這類條件和擬微分算子的關係非常密切。在第四章中，我們使用球調和函數理論去證明變量核奇異積分的 $L^p_w$, $H^p_w-L^p_w$, 和 $h^p_w$ 有界性，其中 $h^p_w$ 代表局部哈弟空間。 第五章考慮一類向量型式的奇異積分算子，我們稱之為帶變量核的分數次 Marcinkiewicz 積分。在一個粗糙核的條件下，我們得到這類算子的 $L^p-L^2$ ($1<p<2$) 有界性。另外，在某種 Dini 型態條件下，得到 $H^p-L^q$ ($ple 1$) 有界性。最後，利用插值定理得到分數次 Marcinkiewicz 積分的 $L^p-L^q$ ($1<p<2$) 有界性。 The main purpose of this thesis is to investigate the weighted boundedness of the singular integral operators with variable kernels. First, we introduce the history of this theory. In Chapter 2, we recall the definitions of $A_p$ weights and the weighted Hardy spaces together with their properties. Next, we separate the smoothness of the variable kernel into two categories. The first is the class of kernels with smoothness in both variables. We adopt Kurtz-Wheeden's method for our estimates to get the $L^p_w$ boundedness of the singular integral operators with kernels in this category. In additional, we also consider the $H^p_w-L^p_w$ boundedness of the operators. These results are given in Chapter 3. The second category is the class of kernels with smoothness in the variable only, which will be discussed in Chapter 4. This category is closely related to the pseudo-differential operators and more appropriate for the work of elliptic differential operators. We use spherical harmonics to decomposition the kernels and obtain the $L^p_w$, $H^p_w-L^p_w$, and $h^p_w$ boundedness of the operators in this category, where $h^p_w$ denotes the local Hardy space introduced by Goldberg and Bui. Finally, we consider a vector-valued version of the singular integral operators, which are called the fractional Marcinkiewicz integrals $mu_{Omega,alpha}$ with variable kernels. Under a rough kernel condition, we get the $L^p-L^2$ ($1<p<2$) boundedness of $mu_{Omega,alpha}$. Then we show that, if the kernel $Omega$ satisfies a class of Dini condition, $mu_{Omega,alpha}$ is boundedness from $H^p$ ($ple 1$) to $L^q$. As a corollary of the above results, we obtain the $L^p-L^q$ ($1<p<2$) boundedness of the fractional Marcinkiewicz integrals. |
