I am currently reading a paper from Del Pino/Dolbeault about optimal constants of GNS inequalities titled Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions.
I am currently on page 4, where the author derives the logarithmic Sobolev inequality from the previous theorem on page 3. There is one step that I don't quite understand though. The author takes the limit when $p\downarrow 1$ in the expression
$\frac{1}{\theta}\log\left(\frac{\lVert w\rVert_{2p}}{\lVert w\rVert_{p+1}}\right)$ and got $\frac{2}{d}\int_{\mathbb{R}^{d}}\left(\frac{w}{\lVert w\rVert_{2}}\right)^{2}\log\left(\frac{w}{\lVert w\rVert_{2}}\right)dx$ using also the approximation $\theta\simeq \frac{d}{4}(p-1)$.
I don't know how to achieve this step. Any help would be appreciated.
I'll assume that $w$ is bounded and compactly supported to not get any integrability problems. Of course, $0^p\log 0$ has to be interpreted as $0$. Let $$ f(p)=\log\left(\frac{\|w\|_{2p}}{\|w\|_{p+1}}\right)=\log\|w\|_{2p}-\log\|w\|_{p+1}. $$ Essentially you want to find $f'(1)$, or, more precisely, the right-sided derivative at $1$. Note that $$ \frac{d^+}{dp}\log\|w\|_p=\frac {d^+} {dp}\left(\frac 1 p\log\int w^p\,dx\right)=-\frac 1{p^2}\log\int w^p\,dx+\frac 1 p\frac{\int w^p\log w\,dx}{\int w^p\,dx}. $$ Thus \begin{align*} \frac{d^+}{dp}\bigg|_{p=1}f(p)&=-\frac 1 2\log\int w^2\,dx+\frac{\int w^2\log w\,dx}{\int w^2\,dx}+\frac 1 4\log\int w^2\,dx-\frac 1 2\frac{\int w^2\log w\,dx}{\int w^2\,dx}\\ &=\frac 1 2\int \frac{w^2}{\|w\|_2^2}\log w\,dx-\frac 1 2\log\|w\|_2\\ &=\frac 1 2\int \frac{w^2}{\|w\|_2^2}\log\frac{w}{\|w\|_2}\,dx. \end{align*} From here it is easy to get the desired limit.