First, I should explain some notation. $A\lesssim B$ denotes the estimate $A\leq C_{\varepsilon}\delta^{-\varepsilon}B.$ If $A$ has polynomial size if $\delta^C\lesssim|A|\lesssim\delta^{-C}$ for some constant $C$. According to author, If I want to prove the following strong-type estimate
$$||X^*f X^*g||_{\frac{n+2}{2n}}\lesssim\delta^{-2\frac{n-2}{n+2}}.$$
It will suffice to show that the following weak-type estimate
$$|\{X^*f X^*g\gtrsim\alpha\}|\lesssim\alpha^{-\frac{n+2}{2n}} \delta^{-\frac{n-2}{n}},\forall\alpha>0.$$
Since the strong-type estimate can be recovered by integrating the above weak-type estimate over all $\alpha$ of polynomial size.
I think the author's meaning might be consider the integral (may be it's not correct)$$\int_{\delta^{C+\varepsilon}}^{\delta^{-C-\varepsilon}}\alpha^{\frac{n+2}{2n}}|\{X^*f X^*g\gtrsim\alpha\}|d\alpha$$
And I have only one idea is that to use the following equation
$$\int|f|^p=p\int_{0}^{\infty}\alpha^{p-1}|\{|f|\geq\alpha\}|d\alpha.$$
However, after calculating, I don't think it's a valid method. So, I want to ask what steps are involved in such a recovery.
The above description is from the paper https://arxiv.org/abs/math/9807163,page 981.