Let $\gamma(x) = (2\pi)^{-1/2}e^{-x^2/2}$ be the standard Gaussian on the real line, and suppose that $f$ is a positive function such that the convolution $f\ast \gamma$ is log-concave, let $G(x)$ be another log-concave function. Is it the case that $(fG)\ast\gamma$ is log-concave?
The motivation to this question is that I am interested in a probability distribution whose density is proportional to $$ p(x) \propto f(x) e^{-x^2/2\sigma^2}, $$ where $f$ has the property described above ($f\ast\gamma$ is log-concave). Since $f$ itself isn't log-concave, one cannot apply Bakry-Emery theory or anything similar to obtain concentration bounds for $p$. However, I still expect to have some amount of concentration, I just don't know how to prove it.