Weak type $A_p$ estimate for bilinear Calderón-Zygmund operators

Abstract

In this paper, we investigate the boundedness of bilinear Calderón-Zygmund operators $T$ from $L^{p_1}\left(w_1\right)\times L^{p_2}\left(w_2\right)$ to $L^{p,\infty }\left(v_{\vec{w}}\right)$ with the stopping time method, where $1/p=1/p_1+1/p_2$ , $1<p_1,p_2<\infty$ and $\vec{w}$ is a multiple $A_{\vec{P}}$ weight. Specifically, we studied the exponent $\alpha$ of $A_{\vec{P}}$ constant in formula $$\Vert T(\vec{f}){\Vert }_{L^{p,\infty }\left(v_{\vec{w}}\right)}⩽C_{m,n,\vec{P},T}[\vec{w}{]}_{A_{\vec{P}}}^{\alpha }{\Vert f_1\Vert }_{L^{p_1}\left(w_1\right)}{\Vert f_2\Vert }_{L^{p_2}\left(w_2\right)}.$$ Surprisingly, we show that when $p⩾\frac{3+\sqrt{5}}{2}$ or $\min \{p_1,p_2\}>4$, the index $\alpha$ in the above equation can be less than $1$, which is different from the linear scenario.