Let $f(x) = \sqrt{1+x^2}^{-\alpha}$, and $g(x) = \sqrt{1+x^2}^{-\beta}$ for some $\alpha, \beta > 0$ satisfying $\alpha + \beta >1$.
Then let $$(f * g)(y) := \int _{\mathbb{R}} \frac{1}{(1+x^2)^{\alpha/2} (1+(x-y)^2)^{\beta/2}} dx~.$$
For all fixed $y$ the above integral converges, thanks to $\alpha+\beta >1$, but we don't assume $f, g \in L^1$, only $fg \in L^1$.
How to prove the decay rate of $f*g$ ?
I suspect it's something like $O(y^{-\min(\alpha,\beta)})$. Thanks
EDIT: Thanks to the calculation of reuns, I also wonder what is the answer in the case where $\alpha = 1$, $\beta >0$ ? ! Thanks