arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2008.13071math.CA
收藏

具有粗糙核的奇异积分的$q$ -变分的定量加权界

Quantitative weighted bounds for the $q$-variation of singular integrals with rough kernels

Yanping Chen, Guixiang Hong, Ji Li

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

本文研究了具有粗糙核的$q$ -变分奇异积分算子的定量加权界。主要结果是为尖锐截断奇异积分本身$$ \|V_q\{T_{Ω,\varepsilon}\}_{\varepsilon>0}\|_{L^p(w)\rightarrow L^p(w)}\leq c_{p,q,n} \|Ω\|_{ L^\infty}(w)_{A_p}^{1+1/q}\{w\}_{A_p},$$的数量$(w)_{A_p}$，$\{w\}_{A_p}$将被召回的介绍;我们不知道这是否是尖锐的，但它是最有名的定量结果为这类运营商，因为当$q=\infty$，它符合最有名的定量界限迪Pilino-Hytönen-李或勒纳。在建立上述估计的过程中，我们得到了几个定量的加权界，这是独立的利益。我们在此强调其中两个。第一个是$$ \|V_q\{ϕ_k\ast T_Ω\}_{k\in\mathbb Z}\|_{L^p(w)\rightarrow L^p(w)}\leq c_{p,q,n} \|Ω\|_{ L^\infty}(w)_{A_p}^{1+1/q}\{w\}_{A_p},$$，其中$ϕ\in C^\infty_c(\mathbb R^n)$是任何非负径向函数的$ϕ_k(x)=\frac1{2^{kn}}ϕ(\frac x{2^k})$，$q=\infty$的锐度是由于Lerner;第二个是$$ \|\mathcal{S}_q\{T_{Ω,\varepsilon}\}_{\varepsilon>0}\|_{L^p(w)\rightarrow L^p(w)}\leq c_{p,q,n} \|Ω\|_{ L^\infty}(w)_{A_p}^{1/q}\{w\}_{A_p},$$，$q=\infty$的锐度来自Hardy-Littlewood极大函数。

英文摘要：

In this paper, we study the quantitative weighted bounds for the $q$-variational singular integral operators with rough kernels. The main result is for the sharp truncated singular integrals itself $$ \|V_q\{T_{Ω,\varepsilon}\}_{\varepsilon>0}\|_{L^p(w)\rightarrow L^p(w)}\leq c_{p,q,n} \|Ω\|_{ L^\infty}(w)_{A_p}^{1+1/q}\{w\}_{A_p},$$ where the quantity $(w)_{A_p}$, $\{w\}_{A_p}$ will be recalled in the introduction; we do not know whether this is sharp, but it is the best known quantitative result for this class of operators, since when $q=\infty$, it coincides with the best known quantitative bounds by Di Pilino--Hytönen--Li or Lerner. In the course of establishing the above estimate, we obtain several quantitative weighted bounds which are of independent interest. We hereby highlight two of them. The first one is $$ \|V_q\{ϕ_k\ast T_Ω\}_{k\in\mathbb Z}\|_{L^p(w)\rightarrow L^p(w)}\leq c_{p,q,n} \|Ω\|_{ L^\infty}(w)_{A_p}^{1+1/q}\{w\}_{A_p},$$ where $ϕ_k(x)=\frac1{2^{kn}}ϕ(\frac x{2^k})$ with $ϕ\in C^\infty_c(\mathbb R^n)$ being any non-negative radial function, and the sharpness for $q=\infty$ is due to Lerner; the second one is $$ \|\mathcal{S}_q\{T_{Ω,\varepsilon}\}_{\varepsilon>0}\|_{L^p(w)\rightarrow L^p(w)}\leq c_{p,q,n} \|Ω\|_{ L^\infty}(w)_{A_p}^{1/q}\{w\}_{A_p},$$ and the sharpness for $q=\infty$ follows from the Hardy--Littlewood maximal function.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑