论文标题
$ Q $ $ Q $变化与粗糙内核的定量加权边界
Quantitative weighted bounds for the $q$-variation of singular integrals with rough kernels
论文作者
论文摘要
在本文中，我们研究了带有粗糙内核的$ q $变量单数集成运算符的定量加权边界。主要的结果是尖锐的截断奇异积分本身 $ \ | v_q \ { l^\ infty}（w）_ {a_p}^{1+1/q} \ {w \} _ {a_p}，$$，其中量化$ $（w）_ {a_p} $，$ \ \ \ \ \ {w \}我们不知道这是否很清晰，但它是这类运营商最著名的定量结果，因为当$ q = \ infty $时，它与di Pilino-hytönen--li或Lerner的最著名的定量界限相吻合。在建立上述估计的过程中，我们获得了几个具有独立利益的定量加权界限。我们在此强调了其中的两个。第一个是 $ \ | v_q \ {ϕ_k \ astt_Ω\} _ {k \ in \ Mathbb z} \ | _ {l^p（w） l^\ infty}（w）_ {a _p}^{1+1/q} \ {w \} _ {a_p}，$ umph，其中$ ϕ_k（x）= \ frac1 = \ frac1 {2^{kn}} ϕ r^n）$是任何非负径向功能，$ q = \ infty $的清晰度归功于勒纳（Lerner）；第二个是 $ \ | \ Mathcal {s} _q \ { l^\ infty}（w）_ {a_p}^{1/q} \ {w \} _ {a_p}，$$和$ q = \ infty $的清晰度来自hardy-little-little-little Wood的最大功能。
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.
