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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.

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