论文标题
关于非标准的粗糙奇异整体操作员的界限
On the boundedness of non-standard rough singular integral operators
论文作者
论文摘要
令$ω$是零度的同质，在单位球体上有一个消失的订单时刻$ \ mathbb {s}^{d-1} $（$ d \ ge 2 $）。在本文中，我们的调查对象是以下粗糙的非标准单数积分运算符$$ t_ {ω，\，a} f（x）= {\ rm p。 $ \ mathbb {r}^d $带有订单的衍生物$ {\ rm bmo}（\ mathbb {r}^d）$。我们表明$ t_ {ω，\，a} $享受端点$ l \ log l $ type估算估计，如果$ l^p $ bughed，则如果$ l^p $，则如果$ log omω\在l（\ log l）^{2}（\ Mathbb {s s}^{d-1-1}）中。这些重新分配基本上改善了Hofmann在条件$ t_ {ω，\，A} $的$ l^p $界面下给出的已知结果{\ rm lip}_α（\ mathbb {s}^{d-1}）$在（0，0，\，1] $中。加权结果在很大程度上取决于在此建立的$ t_ {ω，\，a} $的两个双线性稀疏支配。
Let $Ω$ be homogeneous of degree zero, have vanishing moment of order one on the unit sphere $\mathbb {S}^{d-1}$($d\ge 2$). In this paper, our object of investigation is the following rough non-standard singular integral operator $$T_{Ω,\,A}f(x)={\rm p.\,v.}\int_{\mathbb{R}^d}\frac{Ω(x-y)}{|x-y|^{d+1}}\big(A(x)-A(y)-\nabla A(y)(x-y)\big)f(y){\rm d}y,$$ where $A$ is a function defined on $\mathbb{R}^d$ with derivatives of order one in ${\rm BMO}(\mathbb{R}^d)$. We show that $T_{Ω,\,A}$ enjoys the endpoint $L\log L$ type estimate and is $L^p$ bounded if $Ω\in L(\log L)^{2}(\mathbb{S}^{d-1})$. These resuts essentially improve the previous known results given by Hofmann for the $L^p$ boundedness of $T_{Ω,\,A}$ under the condition $Ω\in L^{q}(\mathbb {S}^{d-1})$ $(q>1)$, Hu and Yang for the endpoint weak $L\log L$ type estimates when $Ω\in {\rm Lip}_α(\mathbb{S}^{d-1})$ for some $α\in (0,\,1]$. Quantitative weighted strong and endpoint weak $L\log L$ type inequalities are proved whenever $Ω\in L^{\infty}(\mathbb {S}^{d-1})$. The analysis of the weighted results relies heavily on two bilinear sparse dominations of $T_{Ω,\,A}$ established herein.
