Without requiring that the integral \begin{equation} \int_1^\infty x(s)\frac{k(s)}{s^2}\,ds \tag{1} \end{equation} be finite, the answer is no. Indeed, then one can take e.g. $k(s)=1/s$ and $x(s)=e^s$.
I don't know whether the condition that the integral in (1) be finite changes the answer.