Limiting weak-type behavior for rough bilinear operators

Let $Omega_1,Omega_2$ be functions of homogeneous of degree $0$ and $vecOmega=(Omega_1,Omega_2)in Llog L(mathbb{S}^{n-1})times Llog L(mathbb{S}^{n-1})$. In this paper, we investigate the limiting weak-type behavior for bilinear maximal function $M_{vecOmega}$ and bilinear singular integral $T_{vecOmega}$ associated with rough kernel $vecOmega$. For all $f,gin L^1(mathbb{R}^n)$, we show that $$lim_{lambdato 0^+}lambda |big{ xinmathbb{R}^n:M_{vecOmega}(f_1,f_2)(x)>lambdabig}|^2 = frac{|Omega_1Omega_2|_{L^{1/2}(mathbb{S}^{n-1})}}{omega_{n-1}^2}prodlimits_{i=1}^2| f_i|_{L^1}$$ and $$lim_{lambdato 0^+}lambda|big{ xinmathbb{R}^n:| T_{vecOmega}(f_1,f_2)(x)|>lambdabig}|^{2} = frac{|Omega_1Omega_2|_{L^{1/2}(mathbb{S}^{n-1})}}{n^2}prodlimits_{i=1}^2| f_i|_{L^1}.$$ As consequences, the lower bounds of weak-type norms of $M_{vecOmega}$ and $T_{vecOmega}$ 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.