real analysis – Lower bound of the decay estimate of a Newtonian potential

I am considering the following Newtonian potential
$$theta(x) = int_{mathbb{R}^3}frac{1}{|x-y|}w(y) dy.$$
Here $w(y) ge 0$ is a nontrivial smooth function in $L^1(mathbb{R}^3)$. Now $theta(x) ge 0$ is well-defined. Can we get the following estimate $$theta(x) ge frac{c}{1+|x|}$$ for some positive constant $c$? I think the non-negativeness of $w$ plays an important role here. If $w$ oscilates and changes sign frequently, I guess $theta$ may decay with a rate faster than $|x|^{-1}$.