My question is pretty simple. Let $A_1, A_2 : L^p(\mu) \to L^{p, w}(\mu)$ be sublinear operators such that there exist constants $C_1, C_2 \in [1, \infty)$ for which \begin{align*} \mu \left( \left\{ x \in X : |A_i f(x)| > t \right\} \right) & \leq \frac{C_i}{t^p} \|f\|_p^p & \left( \forall f \in L^p(\mu), t > 0 \right) \end{align*} for $i = 1, 2$. Here $L^{p, w}$ means weak $L^p$. Does it follow that $A_1 A_2$ satisfies a weak type $(1, 1)$ inequality? Does it even follow that $A_1 A_2$ takes values in weak $L^p$?
The answer is obviously yes if we replace the weak type inequalities with strong type inequalities, but I'm not sure in this circumstance. Thanks in advance for your help.