I have been reading an article and at some point the reach an elliptic PDE, write down the solution using the variation of parameter and then estimate this solution using the integral form of this solution. However, I am not sure to see how they find their estimate.
Let $\varphi$, the solution computed with variation of parameter, defined as $$\varphi(\rho) = Z(\rho) \int_{\rho}^{4R} \frac{dr}{rZ(r)^2}\int_0^rg(s)Z(s)sds,$$ for $R > 0$ and $$Z(\rho) = \frac{2\rho^2}{\rho^2 + 1} \quad \text{and} \quad g(\rho) = \frac{2}{(1 + \rho)^a}$$ for some $a \in (0,3)\backslash \{2\}$. Now from this expression they reach the following estimates, $$|\varphi(\rho)| \le \begin{cases} R^{2 - a} & \text{if } a < 2,\\\\ (1 + \rho)^{2 - a} & \text{if } a > 2. \end{cases}.$$
I've tried many things, but I really don't get how they get such a bound for the solution. Does any of you have an idea of how to get it ?