AI 中文总结
本文研究粗糙极大奇异积分算子及其变差与跳跃算子的加权弱型 (1,1) 有界性，给出了依赖于 A1 和 A∞ 权的定量界，改进了已有结果，并首次建立了粗糙奇异积分的加权变差与跳跃不等式。
AI 中文摘要
本文研究了粗糙极大奇异积分算子以及相应的变差算子和跳跃算子的加权弱型 $(1,1)$ 有界性。我们的第一个结果是极大奇异积分算子 $T_{\Omega}^{*}f(x)=\sup\limits_{\varepsilon>0}|T_{\varepsilon,\Omega}f(x)|=\sup\limits_{\varepsilon>0}\bigg|\int_{|x-y|>\varepsilon} \frac{\Omega(x-y)}{|x-y|^{d}}f(y)dy\bigg|$ 的加权弱型 $(1,1)$ 有界性，其中 $\Omega\in L^\infty(\mathbb{S}^{d-1})$，零次齐次，且满足消失条件。我们证明了 $\\|T_{\Omega}^{*}\\|_{L^1(w)\rightarrow L^{1,\infty}(w)}\lesssim[w]_{A_1}[w]_{A_\infty}\log([w]_{A_\infty}+1)$，其中 $w$ 属于 Muckenhoupt 类 $A_1(\mathbb{R}^d)$。这本质上改进了 Honzík（Int. Math. Res. Not. 2020）和 Bhojak 与 Mohanty（J. Funct. Anal. 2023）的结果。我们的第二个结果是关于 $\{T_{\varepsilon,\Omega}\}_{\varepsilon\in2^{\mathbb{Z}}}$ 和 $\{T_{\varepsilon, \Omega}^{\phi}\}_{\varepsilon\in\mathbb{R}^{+}}$ 的变差算子和跳跃算子的加权弱型 $(1,1)$ 有界性，其中 $\Omega\in L^\infty(\mathbb{S}^{d-1})$，$T_{\epsilon,\Omega}^{\phi}$ 表示粗糙奇异积分的光滑截断。这部分结果是粗糙奇异积分算子的首批加权弱型 $(1,1)$ 变差不等式和跳跃不等式。
英文摘要
In this present paper, we study the weighted weak type $(1,\,1)$ bounds for rough maximal singular integral operators as well as the the corresponding variation and jump operators. Our first result is the weighted weak type $(1,\,1)$ bound for the maximal singular integral $$T_Ω^{*}f(x)=\sup\limits_{\varepsilon>0}|T_{\varepsilon,Ω}f(x)|=\sup\limits_{\varepsilon>0}\bigg|\int_{|x-y|>\varepsilon} \frac{Ω(x-y)}{|x-y|^{d}}f(y)dy\bigg|,$$ where $Ω\in L^\infty(\mathbb{S}^{d-1})$, homogeneous of degree zero, and satisfies the cancellation condition. We show that $$\|T_Ω^{*}\|_{L^1(w)\rightarrow L^{1,\infty}(w)}\lesssim[w]_{A_1}[w]_{A_\infty}\log([w]_{A_\infty}+1),$$ where $w$ belongs to the Muckenhoupt class $A_1(\mathbb{R}^d)$. This represents an essential improvement of a result (Honz\'ık, Inter.Math. Res. Not. 2020) and a result (Bhojak and Mohanty, J. Funct. Anal.2023). Our second one is the weighted weak type $(1,\,1)$ bounds for variation and jump operators corresponding to $\{T_{\varepsilon,Ω}\}_{\varepsilon\in2^{\mathbb{Z}}}$ and $\{T_{\varepsilon, Ω}^ϕ\}_{\varepsilon\in\mathbb{R}^{+}}$ under the condition $Ω\in L^\infty(\mathbb{S}^{d-1})$, where $T_{ε,Ω}^ϕ$ represents a smooth truncation of rough singular integral operator.These results of this part are the first weighted weak type $(1,\,1)$ variation inequalities and jump inequalities for rough singular integrals.
