论文标题
$ l^p（\ mathbb {r}^d）$calderón换向器的边界与粗糙内核
$L^p(\mathbb{R}^d)$ boundedness for the Calderón commutator with rough kernel
论文作者
论文摘要
令$ k \ in \ mathbb {n} $，$ω$是同质零度，可在$ s^{d-1} $上集成，并具有$ k $的消失时刻，$ a $ a $是$ \ mathbb {r}^d $上的函数$ t_ {ω，\，a; k} $是$ d $ -dimensionalcalderón换向器，由$$ t_ {ω，\，a; k} f（x）= {\ rm p。 s^{d-1}} \ int_ {s^{d-1}} |ω（θ）| \ log^β\ big（\ frac {1} {|θ\ cdot面|} \ big） $ \ frac {2β} {2β-1} <p <2β$，$ t_ {ω，\，\，a; \，k} $在$ l^p（\ mathbb {r}^d）上限制。
Let $k\in\mathbb{N}$, $Ω$ be homogeneous of degree zero, integrable on $S^{d-1}$ and have vanishing moment of order $k$, $a$ be a function on $\mathbb{R}^d$ such that $\nabla a\in L^{\infty}(\mathbb{R}^d)$, and $T_{Ω,\,a;k}$ be the $d$-dimensional Calderón commutator defined by $$T_{Ω,\,a;k}f(x)={\rm p.\,v.}\int_{\mathbb{R}^d}\frac{Ω(x-y)}{|x-y|^{d+k}}\big(a(x)-a(y)\big)^kf(y){d}y.$$ In this paper, the authors prove that if $$\sup_{ζ\in S^{d-1}}\int_{S^{d-1}}|Ω(θ)|\log ^β \big(\frac{1}{|θ\cdotζ|}\big)dθ<\infty,$$ with $β\in(1,\,\infty]$, then for $\frac{2β}{2β-1}<p<2β$, $T_{Ω,\,a;\,k}$ is bounded on $L^p(\mathbb{R}^d)$.
