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

We investigate the boundedness of bilinear Calderón–Zygmund operators $T$ from $L^{p_1}(w_1) \times L^{p_2}(w_2)$ to $L^{p,\infty}(v_{\vec w})$ with the stopping time method, where $1 / p = 1 / p_1 + 1 / p_2$ , $1 \lt p_1, p_2 \lt \infty $ and $\vec w$ is a multiple $A_{\vec P}$ weight. Specifically, we study the exponent $\alpha $ of the $A_{\vec P}$ constant in the formula $$\|T(\vec f)\|_{L^{p,\infty}(v_{\vec w})} \leq C_{m, n, \vec P, T}[\vec w]_{A_{\vec P}}^{\alpha }\|f_1\|_{L^{p_1}(w_1)}\|f_2\|_{L^{p_2}(w_2)}.$$ Surprisingly, we show that when $p \geqslant \frac {3+\sqrt 5}{2}$ or $\min\,\{p_1,p_2\} \gt 4$, the exponent $\alpha $ can be less than $1$, which is different from the linear scenario.