## ρ-Variation for Singular Integral Operators with Variable Kernels

Gong Zhenbing,, Chen Yanping,, Tao Wenyu,

 基金资助: 国家自然科学基金.  11871096国家自然科学基金.  11471033

 Fund supported: the NSFC.  11871096the NSFC.  11471033

Abstract

In this paper, we will prove that the variation of singular integral operators with rough variable kernels are bounded on $L^{2}({\mathbb S} ^{n})$ if $\Omega\in L^{\infty}({\mathbb R} ^{n})\times L^{q}({\mathbb S}^{n-1})$ for $q>2(n-1)/n$, and $n\geq2$. Moreover, we can also obtain the weighted variational inequalities for singular integral operators with smooth variable kernels. Finally, we extend the result to the Morrey spaces.

Keywords： Variational inequalities ; Variable kernel ; Singular integral ; Ap-weight ; Morrey space

Gong Zhenbing, Chen Yanping, Tao Wenyu. ρ-Variation for Singular Integral Operators with Variable Kernels. Acta Mathematica Scientia[J], 2020, 40(6): 1446-1460 doi:

## 1 引言

${\mathbb S}^{n-1} = \{x \in {{\Bbb R}} ^{n}:|x| = 1\} $${{\Bbb R}} ^{n}\, (n\geq2) 上的单位球面, {\rm d}\sigma 表示 {\mathbb S}^{n-1} 上的Lebesgue测度.设 {{\Bbb R}} ^{n}\times{\mathbb S}^{n-1} 上的零阶齐次函数 \Omega(x, z') \in L^{\infty}({{\Bbb R}} ^{n}) \times L^{q}({\mathbb S}^{n-1})(q\geq 1) 是指 $$\Omega(x, \lambda z) = \Omega(x, z)\quad \mbox{对每个 } x, z \in {{\Bbb R}} ^{n} \mbox{ 且 } \lambda>0,$$ $$\|\Omega\|_{L^{\infty}({{\Bbb R}} ^{n}) \times L^{q}({\mathbb S}^{n-1})}: = \sup\limits_{x \in {{\Bbb R}} ^{n}}\bigg(\int_{{\mathbb S}^{n-1}}|\Omega(x, z')|^{q} {\rm d}\sigma(z')\bigg)^{1 / q}<\infty,$$ 其中 z' = z /|z|,$$ z \in {{\Bbb R}} ^{n} \backslash\{0\} .$

$\begin{eqnarray} T_{\Omega, \varepsilon} f(x) = \int_{|x-y|>\varepsilon}\frac{\Omega(x, x-y)}{|x-y|^n}f(y){\rm d}y, \end{eqnarray}$

$$$\int_{{\mathbb S}^{n-1}} \Omega(x, y') {\rm d}\sigma(y^{\prime}) = 0,$$$

$$$T_\Omega f(x) = \lim\limits_{\varepsilon\rightarrow0^+}T_{\Omega, \, \varepsilon} f(x), \ a.e. \ x\in{{\Bbb R}} ^n.$$$

${\cal T}_\Omega$表示算子族$\{T_{\Omega, \varepsilon}\}_{\varepsilon>0}$.众所周知, ${\cal T}_\Omega$的变差不等式在学习带变量核奇异积分算子簇的几乎处处收敛性问题中也起到了至关重要的作用.受到定理A的启发, 一个有趣的问题是, ${\cal T}_\Omega$的变差是否也能得到相同的结果.该文首先对这个问题做出了一个肯定的回答, 并建立了带粗糙核${\cal T}_\Omega$的变差函数是$L^{2}({{\Bbb R}} ^{n})$有界的.

$\rho\geq1$, 并且${\mathfrak{a}} = \{a_{t}:t>0\}$是一个复数族, 复数族${\mathfrak{a}}$$\rho$ -变差范数定义为

$$$\|{\mathfrak{a}}\|_{L^{\infty}({{\Bbb R}} )}: = \sup\limits_{t \in {{\Bbb R}} }|a_{t}| \leq\|{\mathfrak{a}}\|_{V_{\rho}}+|a_{t_{0}}| \quad \rho \geq 1.$$$

${\cal F} = \{F_{t}\}_{ t>0}$是定义在${{\Bbb R}} ^{n}$上的Lebesgue可测函数族.通过复数族的$\rho$ -变差范数的定义, 可以定义函数族${\cal F}$的强$\rho$ -变差函数$V_{\rho}({\cal F})$.固定$x\in{{\Bbb R}} ^{n}$, 函数族${\cal F}$