论文标题
限制双线性双线操作员的弱型行为
Limiting weak-type behavior for rough bilinear operators
论文作者
论文摘要
令$ω_1，ω_2$为均值$ 0 $和$ \ \vecΩ=（ω_1，ω_2）\ in L \ log l（\ Mathbb {s}^{n-1}）\ times log log l \ log l（\ log l（\ mathbb {s} s}^n-1}）$。在本文中，我们研究了双线性最大函数的限制弱型行为$ m _ {\vecΩ} $和双线性单数积分$ t _ {\vecΩ} $与rugh-bernel $ \vecΩ相关的。对于所有$ f，g \ in l^1（\ mathbb {r}^n）$，我们表明，$ \ lim_ {λ\ to 0^+}λ| \ big \ big \ {x \ in \ mathbb {r} \frac{\|Ω_1Ω_2\|_{L^{1/2}(\mathbb{S}^{n-1})}}{ω_{n-1}^2}\prod\limits_{i=1}^2\| f_i \ | _ {l^1} $$和$$ \ lim_ {λ\ to 0^+}λ| \ big \ big \ {x \ in \ Mathbb {r}^n：| | t _ {\vecΩ}（f_1，f_2）（x）|>λ\ big \} |^{2} = \ frac {\ |ω_1Ω_2\ | _ {l^{1/2}（\ Mathbb {s}}^{n-1}}}}}}} {n^2} \ prod \ limits_ {i = 1}^2 \ | f_i \ | _ {l^1}。$$作为后果，获得了$ m _ {\vecΩ} $的弱型规范的下限和$ t _ {\vecΩ} $。即使在线性情况下，这些结果也是新的。还讨论了粗糙双线性分数最大函数和分数积分操作员的相应结果。
Let $Ω_1,Ω_2$ be functions of homogeneous of degree $0$ and $\vecΩ=(Ω_1,Ω_2)\in L\log L(\mathbb{S}^{n-1})\times L\log L(\mathbb{S}^{n-1})$. In this paper, we investigate the limiting weak-type behavior for bilinear maximal function $M_{\vecΩ}$ and bilinear singular integral $T_{\vecΩ}$ associated with rough kernel $\vecΩ$. For all $f,g\in L^1(\mathbb{R}^n)$, we show that $$\lim_{λ\to 0^+}λ|\big\{ x\in\mathbb{R}^n:M_{\vecΩ}(f_1,f_2)(x)>λ\big\}|^2 = \frac{\|Ω_1Ω_2\|_{L^{1/2}(\mathbb{S}^{n-1})}}{ω_{n-1}^2}\prod\limits_{i=1}^2\| f_i\|_{L^1}$$ and $$\lim_{λ\to 0^+}λ|\big\{ x\in\mathbb{R}^n:| T_{\vecΩ}(f_1,f_2)(x)|>λ\big\}|^{2} = \frac{\|Ω_1Ω_2\|_{L^{1/2}(\mathbb{S}^{n-1})}}{n^2}\prod\limits_{i=1}^2\| f_i\|_{L^1}.$$ As consequences, the lower bounds of weak-type norms of $M_{\vecΩ}$ and $T_{\vecΩ}$ are obtained. These results are new even in the linear case. The corresponding results for rough bilinear fractional maximal function and fractional integral operator are also discussed.
