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 $$|x|^{-1}$$.