# Product of distributions satisfying log-sobolev inequality

Let $$f,g\in C^\infty(\mathbb{R})$$ be two smooth positive functions satisfying $$\int f = \int g = 1$$. Suppose that both $$f$$ and $$g$$ satisfy the log-Sobolev inequality (LSI) with constant $$C$$, so that $$\int \phi^2 \log(\phi^2) f(x)\,dx \leq C \int |\phi'|^2 f(x)\,dx$$ for every $$\phi$$ satisfying $$\int \phi^2(x)f(x)\,dx = 1$$, and the same holds with $$f$$ replaces by $$g$$.

If $$\int f(x)g(x)\,dx = Z > 0$$, does it follow that the density $$h(x) = Z^{-1} f(x)g(x)$$ also satisfies LSI with the same constant $$C$$?

I would be equally interested in a counterexample.