Limiting weak-type behavior for rough bilinear operators

Abstract

Let $\Omega_1,\Omega_2$ be functions of homogeneous of degree $0$ and $\vec\Omega=(\Omega_1,\Omega_2)\in L\log L(\mathbb{S}^{n-1})\times L\log L(\mathbb{S}^{n-1})$. In this paper, we investigate the limiting weak-type behavior for bilinear maximal function $M_{\vec\Omega}$ and bilinear singular integral $T_{\vec\Omega}$ associated with rough kernel $\vec\Omega$. For all $f,g\in L^1(\mathbb{R}^n)$, we show that $$\lim_{\lambda\to 0^+}\lambda |\big\{ x\in\mathbb{R}^n:M_{\vec\Omega}(f_1,f_2)(x)>\lambda\big\}|^2 = \frac{\|\Omega_1\Omega_2\|_{L^{1/2}(\mathbb{S}^{n-1})}}{\omega_{n-1}^2}\prod\limits_{i=1}^2\| f_i\|_{L^1}$$ and $$\lim_{\lambda\to 0^+}\lambda|\big\{ x\in\mathbb{R}^n:| T_{\vec\Omega}(f_1,f_2)(x)|>\lambda\big\}|^{2} = \frac{\|\Omega_1\Omega_2\|_{L^{1/2}(\mathbb{S}^{n-1})}}{n^2}\prod\limits_{i=1}^2\| f_i\|_{L^1}.$$ As consequences, the lower bounds of weak-type norms of $M_{\vec\Omega}$ and $T_{\vec\Omega}$ 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.