论文标题
在非本地方程中出现的Riesz-Transform类型运算符的Calderon-Zygmund属性上
On the Calderon-Zygmund property of Riesz-transform type operators arising in nonlocal equations
论文作者
论文摘要
我们表明操作员\ [ t_ {k，s_1，s_2} f（z）：= \ int _ {\ mathbb {r}^n} a_ {k，s_1，s_1，s_1，s_1，z_1，z_1，z_2）f（z_2）f（z_2）\，dz_2 \]是Calderon-Zygmund opertortortortortor。在l^\ infty（\ mathbb {r}^n \ times \ times \ mathbb {r}^n）$和$ s，s_1，s_1，s_2 \ in（0,1）$带有$ s_1+s_2 = 2s $我们有\ [我们有\ [我们有\ [我们有\ [0,1）$的$ k \ in。 a_ {k，s_1，s_2}（z_1，z_2）= \ int _ {\ mathbb {\ mathbb {r}^n} \ int _ {\ mathbb {\ mathbb {r}^n} \ frac {k（k（y）） \ right）\，\ left（| x-z_2 |^{s_2-n} - | y-z_2 |^{s_2-n} \ right）} {| x-y |^{n+2s}}} \，dx \，dx \，dy。 \]该操作员是由Mengesha-Schikorra-Yeepo最近的工作激励的，它看起来像是Riesz的类似物，用于方程式\ [ \ int _ {\ Mathbb {r}^n} \ int _ {\ Mathbb {r}^n} \ frac {k（x，y）（u（x）-u（x）-u（x）-u（y（y））\，（φ（x） - x）-（x）-（x）-（x）-（y））}} \]
We show that the operator \[ T_{K,s_1,s_2}f(z) := \int_{\mathbb{R}^n} A_{K,s_1,s_2}(z_1,z_2) f(z_2)\, dz_2 \] is a Calderon-Zygmund operator. Here for $K \in L^\infty(\mathbb{R}^n \times \mathbb{R}^n)$, and $s,s_1,s_2 \in (0,1)$ with $s_1+s_2 = 2s$ we have \[ A_{K,s_1,s_2}(z_1,z_2) = \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{K(x,y) \left (|x-z_1|^{s_1-n} -|y-z_1|^{s_1-n} \right )\, \left (|x-z_2|^{s_2-n} -|y-z_2|^{s_2-n}\right )}{|x-y|^{n+2s}}\, dx\, dy. \] This operator is motivated by the recent work by Mengesha-Schikorra-Yeepo where it appeared as analogue of the Riesz transforms for the equation \[ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{K(x,y) (u(x)-u(y))\, (φ(x)-φ(y))}{|x-y|^{n+2s}}\, dx\, dy = f[φ]. \]
