Another approach to the equality case of the sharp logarithmic Sobolev inequality

In this note, we characterize the equality case of the sharp $L^2$-Euclidean logarithmic Sobolev inequality with monomial weights, exploiting the idea by Bobkov and Ledoux \cite{Bob}. Our approach is new even in the unweighted case. Also, we show that the same strategy yields the equality case of the sharp $L^p$-Euclidean logarithmic Sobolev inequality for $1<p<\mathrm{\infty }$ with arbitrary norm on ${\mathbb{R}}^n$, if the inequality is unweighted.